Mathematics (2 unit) – Sequences and Series

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Sequences and Series

In this topic we shall study sequences and series, and their properties. Firstly we define the terms sequence and series.

Sequence: Any mathematical progression of numbers, following a pattern. (e.g. Latex formula and Latex formula )

Series: The sum of a finite or infinite sequence of terms. (e.g. Latex formula and Latex formula )

In general, sequences follow some rule. The rule is called the general term. For the sequence Latex formula the general term is Latex formula where Latex formula represents the numbers Latex formula. The general term, or rule may be called the Latex formulath term.

Example 1

Write down the first three terms for the sequence given by the rule Latex formula.

Solution 1

We have that,

 

Latex formula

 

Latex formula

 

Latex formula

 

Hence we have that the first three terms Latex formula are given by Latex formula and Latex formula respectively.

Note: The expression Latex formula is in fact a short hand convention for Latex formula. This is because a sequence is in fact simply the range of a function with a domain equal to the natural numbers. That is, the general rule, is simply a function Latex formulawhich converts natural numbers into a sequence.

Arithmetic Sequences

A special family of sequences called arithmetic sequences (or arithmetic progressions abbreviated to AP), are of particular interest to us. In an arithmetic progression, ensuing terms follow the pattern of differing by the same amount. For example, consider the arithmetic progression Latex formula all the adjacent terms of which differ by Latex formula.

Example 2

Show that Latex formula forms an arithmetic progression.

Solution 2

For three successive terms to form an arithmetic progression, we simply require that the difference between the third and second term be equal to the difference between the second and first term. That is, each successive pair, differ by the same amount.

So we have that,

 

Latex formula And Latex formula

 

Hence as the difference is constant, then it follows that the terms form an arithmetic progression.

As a convention, we denote the common difference between successive terms of an AP by Latex formula. That is,

 

Latex formula

 

We shall now find the general term describing the Latex formula th term of any arithmetic progression. Suppose that the first term of an arithmetic progression is Latex formula and that the common difference is given by Latex formula. Then we have the following table.

Latex formula Latex formula Latex formula Latex formula Latex formula
Latex formula Latex formula Latex formula Latex formula Latex formula

By observation of the general pattern, we can observe that the Latex formula th term of an arithmetic sequence is given by;

Latex formula

where Latex formula is the first term, Latex formula is the common difference and Latex formula is the term number.

We shall now consider some examples to illustrate use of this formula.

Example 3

Find Latex formula of Latex formula.

Solution 3

So we have that Latex formula and Latex formula. Hence we have that the general term is given by,

 

Latex formula

 

So, we have that,

 

Latex formula

 

Example 4

Find the term number that is equal to Latex formula in the sequence Latex formula.

Solution 4

Firstly, we find the general term. We have that Latex formula and Latex formula. Hence we have that

 

Latex formula

 

So we must set the general term to Latex formula and solve the resulting equation in Latex formula to find the term number.

 

Latex formula

 

Hence we have that the Latex formula rd term is equal to Latex formula.

 

Example 5

The Latex formula rd term of an AP is Latex formula and the Latex formula st term is Latex formula. Find the first term and the common difference of this sequence.

Solution 5

We firstly have that the general term of an AP is given by Latex formula. We have that,

 

Latex formula

 

Latex formula

 

So we have,

 

Latex formula

 

And

 

Latex formula

 

Now, Latex formula gives us,

 

Latex formula

 

Now, substituting this back into Latex formula gives,

 

Latex formula

 

Thus we have that the first term is Latex formula and the common difference is Latex formula for this sequence.

 

Arithmetic Series

We now shall look at arithmetic series (the summation of an arithmetic progression). We denote the sum up to Latex formula terms for any series as Latex formula.

Now, suppose we are adding Latex formula terms from an arithmetic progression to obtain a sum. That is,

 

Latex formula

 

Now, we consider Latex formula written in the reverse order with terms lining up vertically with Latex formula written in correct order.

Latex formula Latex formula Latex formula Latex formula Latex formula Latex formula
Latex formula Latex formula Latex formula Latex formula Latex formula Latex formula

Now, adding the two series with terms lining vertically being added gives,

 

Latex formula

 

Now, on the RHS there exists Latex formula terms. So we have,

 

Latex formula
Latex formula

which is the formula for the summation of an AP. We may further simplify the formula;

 

Latex formula

 

Now, since there exists Latex formula terms in the series, we thus have that Latex formula where Latex formula is the last term. Hence we have that,

Latex formula

where Latex formula is the first term, Latex formula is the number of terms in the series, Latex formula is the last term Latex formula and Latex formula is the common difference.

Note: The second form of the summation formula for an AP is only used if the last term is known. Otherwise we simply use the standard form which is the first.

Example 6

What is the sum of the first 80 terms of the sequence Latex formula?

Solution 6

We have that Latex formula, Latex formula and Latex formula. We are required to calculate Latex formula.

 

Latex formula

 

Latex formula

 

Thus the sum of the first Latex formula terms is Latex formula.

Now, in some questions, students are given the summation formula, and are required to find the general term of the sequence being summed. This not only applies to AP’s but to all series. The method used to find the general term given the summation formula for Latex formulaterms is,

Latex formula

Now, this formula is obviously true since,

 

Latex formula

 

Latex formula is obtained by substituting Latex formula whenever Latex formula occurs in the formula for Latex formula.

Consider the below example to illustrate the method used throughout such questions.

Example 7

Find the general term for the sequence being summed if the formula for the sum of the first Latex formula terms of this sequence is given by Latex formula, and hence find the Latex formula rd term of the sequence.

Solution 7

Latex formula

 

Now,

 

Latex formula

 

Latex formula

 

Latex formula

 

Hence we have that,

 

Latex formula

 

That is, the general term is Latex formula and the Latex formula rd term is given by . Latex formula

 

Sigma Notation

Here we introduce a notation for the summation of a series. The Greek letter Latex formula (Sigma) is used to denote the summing of terms. With the summation formula, we simply place the values of Latex formula from the lowest value (shown at the bottom) to the highest value (shown at the top), into the formula, incrementing the number by Latex formula, and then simply sum the terms together. Examples are the best way to introduce this notation.

Example 8

Find the value of

 

Latex formula

 

Solution 8

Latex formula

 

Latex formula

 

Latex formula

 

Latex formula

 

Geometric Sequences

Another special sequence of numbers is called a geometric sequence or geometric progression (GP).

These sequences have the property that each succeeding term is a constant multiple of the previous term. This constant multiple is called the common ratio and is denoted Latex formula (for ratio). An example of such a series is Latex formula where the common ratio is Latex formula.

For a sequence of numbers to be a geometric progression, the ratio of any two adjacent terms (succeeding term over preceding term) must be equal to the common ratio. In other words,

 

Latex formula

 

For all natural numbers Latex formula.

Example 9

Show that the sequence Latex formula is a geometric progression.

Solution 9

For the sequence to be a geometric progression, we must simply show that ordered adjacent terms share a common ratio.

 

Latex formula

 

Latex formula

 

Hence as the ratio of the third term to the second is equal to the ratio of the second term to the first, then it follows that Latex formula is a GP.

We shall now find the Latex formula th term of a geometric sequence. Suppose that a geometric sequence has first term Latex formula and common ratio Latex formula. We have that,

Latex formula Latex formula Latex formula Latex formula Latex formula
Latex formula Latex formula Latex formula Latex formula Latex formula

It is obvious from that pattern that the Latex formula th term of a geometric sequence is,

Latex formula

where Latex formula is the term number, Latex formula is the first term and Latex formula is the common ratio.

Consider the below examples to illustrate the use of the formula.

Example 10

Find Latex formula of the following GP;

 

Latex formula

Solution 10

We have that Latex formula and Latex formula. Hence we have that,

 

Latex formula

 

and consequently,

 

Latex formula

 

Example 11

Find the number of terms in the following GP;

 

Latex formula.

Solution 11

So we have Latex formula, Latex formula. Hence we have that,

 

Latex formula

 

Now, we have that the last term of the sequence is Latex formula. Equating this t Latex formula and solving the resultant equation for Latex formula gives;

 

Latex formula

 

Latex formula

 

Now, we must take logarithms of both sides to solve for Latex formula.

 

Latex formula

 

Now, we have that,

 

Latex formula

 

By the rule Latex formula.

Hence we have that,

 

Latex formula

 

Hence we have that,

 

Latex formula

 

Hence there are Latex formula terms in the sequence.

 

Example 12

In a certain geometric progression Latex formula and Latex formula. Find the general term.

Solution 12

Assuming the general form for the Latex formula th term, we have that;

 

Latex formula

 

Now,

 

Latex formula

 

And

 

Latex formula

 

Hence we have two equations;

 

Latex formula

 

Latex formula

 

Now Latex formula gives,

 

Latex formula

 

Latex formula

 

Substituting back into Latex formula gives,

 

Latex formula

 

Hence we have that the general term is given by,

 

Latex formula

Geometric Series

If we are to sum a geometric progression, we obtain a geometric series. We shall denote the sum to terms Latex formula as Latex formula. Hence we have that,

 

Latex formula

 

To obtain a summation formula for geometric series, we must firstly perform the operation, Latex formula.

 

Latex formula

 

That is,

 

Latex formula

 

Latex formula

 

Upon cancelation of terms. Now, dividing through by Latex formula and factoring Latex formula gives,

Latex formula

Provided that Latex formula. By removing a factor of Latex formula from top and bottom we obtain,

Latex formula

with Latex formula.

Obviously both formulae are equivalent and it doesn’t matter which formula one uses, however, the use of the first is recommended when Latex formula and the use of the second is recommended when Latex formula to avoid difficulties with the negatives.

Example 13

Find the sum to 12 terms of the geometric sequence Latex formula.

Solution 13

We have that Latex formula and that Latex formula. Also we are given that Latex formula.

Hence we have,

 

Latex formula

Limiting Sum of a Geometric Series

In the case where we have that Latex formula (i.e Latex formula ) we have the following situation.

As Latex formula, Latex formula since Latex formula. Hence we have that,

 

Latex formula

 

Latex formula

 

Latex formula

 

That is, the sum to an infinite number of terms, of a geometric series with common ratio between Latex formula and Latex formula is a finite value. We denote this by Latex formula. Hence we have the formula,

Latex formula

when Latex formula.

Note: If Latex formula then the limiting sum of the geometric series does not exist and the formula does not hold.

We shall now examine some of the uses of this formula.

Example 14

Does the limiting sum of the geometric series Latex formula exist? If so, find its value.

Solution 14

Firstly, the only condition required for the limiting sum to exist is that Latex formula. We have that,

 

Latex formula

 

Hence the limiting sum does exist.

Now, we have that Latex formula and Latex formula. Applying the formula for infinite geometric series gives,

 

Latex formula

Questions involving limiting sums may at times involve expressing repeating decimals as fractions. The technique for such questions is quite straightforward. Observe the example below.

Example 15

Express Latex formula as a fraction, using geometric series.

Solution 15

Firstly we write Latex formula as a decimal.

 

Latex formula

 

Latex formula

 

Now, we have that this repeating decimal can be represented in terms of fractions as

 

Latex formula

 

One can see that the bracketed segment is a geometric series with Latex formula and Latex formula.

Hence we have that,

 

Latex formula

 

That is,

 

Latex formula

Exam Style Questions

Here we consider three exam style questions, typical of assessments and exams.

Example 16

The sum of the first five terms of an arithmetic series is four times the fourth term. Also, the sum of the fifth and sixth terms is 65. Find the sum of the first 15 terms of the series.

Solution 16

So firstly, the written information must be translated into symbolic mathematics. We have that,

 

Latex formula

 

and

 

Latex formula

 

Now, we know that Latex formula since the sequence is an AP, and also that,

 

Latex formula

 

Using these formulae gives,

 

Latex formula

 

and

 

Latex formula

 

Hence we have the equations,

 

Latex formula Latex formula

 

Now, substituting Latex formula into Latex formula gives,

 

Latex formula

 

and hence we have that

 

Latex formula

 

Now, we are required to find Latex formula.

 

Latex formula

 

Hence the sum of the first 15 terms is Latex formula.

 

Example 17

Find the number which added to each of Latex formula and Latex formula will give a set of three numbers in geometric progression.

Solution 17

So, let this number be Latex formula. So we have that Latex formula must form a geometric progression.

So we require that,

 

Latex formula

 

Since ordered adjacent terms must have the same common ratio.

 

Latex formula

 

Upon cross multiplying.

 

Latex formula

 

That is,

 

Latex formula

 

Hence the number is Latex formula.

 

Example 18

a) Show that

 

Latex formula

 

b) Hence find value of,

 

Latex formula

Solution 18

a) The series forms an Arithmetic Series, with Latex formula and Latex formula. Now, to find the number of terms, we simply need to find the Latex formula th term of the sequence and equate this to the last term, upon which we may solve for Latex formula and find the term number. So we have that,

 

Latex formula

 

So, equating this to Latex formula gives,

 

Latex formula

 

Latex formula

 

We can now use the formula for summing an AP. In this case we shall use the last term form of the formula.

 

Latex formula

 

b) Now, to sum the series,

 

Latex formula

 

We firstly notice that this can be broken up into the series from (a) and another geometric series.

 

Latex formula

 

Latex formula

 

Now, there are 16 terms for the series from (a) and hence there are 16 terms for the geometric series as well. Now, we can see that for the geometric series, Latex formula and Latex formula. Hence we have that the sum is equal to,

 

Latex formula

 

Using (a) and the summation formula for a GP.

Simplifying gives,

 

Latex formula

 

Hence we have that,

 

Latex formula

Series Applications

A practical application of arithmetic and geometric series arises within financial mathematics. Within the Advanced Mathematics course there are 2 main types of questions related to financial mathematics; Superannuation and Time Payments.

Superannuation: A fixed amount is invested each period over a number of years.

Time Payment: A loan is taken, and the repayments are made until the total amount owing is zero. Interest is charged on the amount owing periodically.

It is not necessary to know the details of these financial terms to be able to solve the mathematical problems correctly. More importantly, students should be able to convert a worded problem into symbolic mathematics.

Other applications also arise including physical problems (construction problems), compound interest and wage increase.

Example 19

A pile of sand dumped at the start of a long road has to be distributed in truckloads at Latex formula metre intervals along the road. If there are Latex formula truckloads of sand in the pile, calculate how far in total the truck has to travel in order to deliver the sand.

Solution 19

Observe the diagram below depicting the situation.

depicting diagram of arithmetic and geometric series

For every truckload the truck must travel back and forth along the same distance. This doubles every distance travelled.

It is obvious that the total distance travelled is an arithmetic series, since each time the truck goes to the next point to dump the sand, the distance constantly increases by Latex formula m from the previous point.

Now, the first distance travelled is Latex formula since the pile is put Latex formula m from where the big pile is. This becomes our value of Latex formula. The common difference between distances travelled is Latex formula m since every time the truck travels back and forth, an extra 1600m is added onto the previous distance travelled. So, as there are 15 piles, then we have that the total distance Latex formula is given by,

 

Latex formula

 

Hence the total distance travelled by the truck is Latex formula km.

 

Example 20

Find the accumulated amount after Latex formula years, if Latex formula dollars is compounded annually at Latex formula per annum.

Solution 20

So, let Latex formula be the amount accumulated after Latex formula years. We have that

 

Latex formula

 

Since after at the start, no interest is to be paid on the principle. Now, after the first year, Latex formula interest is paid on the principle and added to the total. So we have that Latex formula is equal to,

 

Latex formula

 

Now, after 2 years, Latex formula interest is paid on the total amount accumulated and then added to the total. That is,

 

Latex formula

 

We can now see a general pattern here, and may extrapolate the accumulated amount after Latex formula years as:

 

Latex formula

 

By comparing the changing factors in Latex formula and Latex formula.

Hence the total accumulated amount after Latex formula years is

 

Latex formula

 

The above example formulates the compound interest formula. The next question is an example of a time payment question.

Example 21

A family loans Latex formula to buy a house. The interest rate is at a constant 9% p.a., and the family wishes to repay the money over 20 years. Find the monthly repayments required for this to occur.

Solution 21

We firstly note that after 20 years there occurs Latex formula years. Letting Latex formula be the amount owing after Latex formula months, we require Latex formula. Also, the interest rate is in terms of years. We require the calculation to be in terms of months. Hence the interest rate is now

Latex formula.

 

We note that,

 

Latex formula

And after one month we have interest added on to the total (so we multiply the total amount owing by Latex formula where Latex formula is the interest rate), after which the monthly repayment Latex formula made. That is,

 

Latex formula

 

Now, after the second month, interest is charged on the previous months final balance, after which another monthly repayment is made.

 

Latex formula Latex formula

Upon substituting the expression for Latex formula in terms of Latex formula into the expression.

Now, we have that after the third month, interest is charged on the previous months balance, after which another monthly repayment is made.

 

Latex formula

 

Latex formula

 

From here we can see a general pattern for Latex formula. Extrapolating this gives,

 

Latex formula

 

Now, using the summation formula for a GP on the second expression gives,

 

Latex formula

 

Now, we require Latex formula. Using this, and solving for Latex formula gives,

 

Latex formula

 

Latex formula

 

i.e. Latex formula$Latex formula

Hence the monthly repayments are $Latex formula.

The next question is an example of a superannuation problem.

Example 22

Nam’s father paid $Latex formula into an account on the day Nam was born. After that, he paid $Latex formula into the account on Nam’s birthday until Nam’s 18th birthday (He does not pay on Nam’s 18th birthday). If the account accrued interest Latex formula at per annum compounded monthly, calculate how much Nam would receive on his 18th birthday.

Solution 22

Firstly, we let Latex formula be the amount accumulated after Latex formula years.

Latex formula

is obviously true. Now, after the first year, this amount is compounded monthly at Latex formula per annum. The monthly interest rate is Latex formula and the total number of periods in this time is Latex formula. Also, at the end of the first year, $Latex formula is added into the account. Applying the compound interest formula gives and adding $Latex formula gives,

 

Latex formula

 

Now, at the end of the second year, Latex formula per annum interest is paid on the accumulated amount, and an extra $Latex formula is added into the account. We have;

 

Latex formula

 

Substituting the previous expression for Latex formula in terms of Latex formula gives;

 

Latex formula

 

Latex formula

 

We can now extrapolate the Latex formula th term as,

 

Latex formula

 

Now the expression contained within brackets is obviously a geometric series with Latex formulaLatex formula and the number of terms being Latex formula. Hence we have that,

 

Latex formula

 

Now, to find the amount accumulated after the Latex formula th birthday, we simply substitute Latex formula into the above expression for Latex formula.

 

Latex formula = $Latex formula

 

Now, since Nam’s father does not pay on the 18th birthday, and the above formula assumes he does, we must subtract $Latex formula only. We do not need to consider any interest to subtract since the $Latex formula is added at the time of the 18th birthday according to the above formula, thus not allowing it to accrue any interest of its own.

 

$Latex formula – $Latex formula = $Latex formula

 

Hence we have that Nam will obtain Latex formula dollars on his Latex formula th birthday.

It can be seen that both time payment and superannuation problems can be solved using the same techniques. Simply formulate an expression for by Latex formula writing the first few terms down, use the geometric series formulae to simplify the expression, and calculate what is required.

Note: It is important that you do not round off numbers early (especially the modified interest rate), as this will lead to very large differences later on in the calculation. Try to make use of the memory function of your calculator.

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