Arithmetic and Geometric Progressions

If you have the sequence 2, 8, 14, 20, 26, then each term is 6 more than the previous term. This is an example of an arithmetic progression (AP) and the constant value that defines the difference between any two consecutive terms is called the common difference.

If an arithmetic difference has a first term a and a common difference of d, then we can write

a, (a + d), (a + 2d),... {a + (n-1) d}

where the nth term = a + (n−1)d

Sum of Arithmetic series

The sum of an arithmetic series of n terms is found by making n/2 pairs each with the value of the sum of the first and last term. (Try this with the sum of the first 10 integers, by making 5 pairs of 11.)

This gives us the formula:

Sum

where a = first term and l = last term.

As the last term is the nth term = a + (n − 1)d we can rewrite this as:

Sum

(Use the first formula if you know the first and last terms; use the second if you know the first term and the common difference.)

If you have a sequence such as: 81, 27, 9, 3, 1, 1/3, 1/9,... then each term is one third of the term before.

This can be written as 81, 81(1/3), 81(1/3)2, 81(1/3)3, 81(1/3)4,...

It is an example of a Geometric Progression (GP) where the each term is a multiple of the previous one. The multiplying factor is called the common ratio.

So a GP with a first term a and a common ratio r with n terms, can be stated as

a, ar, ar2, ar3, ar4...arn-1 , where the nth term = arn-1

Example:

In the sequence, 400, 200, 100, 50,... find the 8th term.

a = 400, r = 0.5 and so the 8th term = 400 × 0.57 = 3.125

Note: To find which term has a certain value you will need to use logarithms.

Example:

In the sequence, 2, 6, 18, 54 ... which is the first term to exceed 1,000,000?

a = 2, r = 3.

2 × 3n-1 > 1,000,000

3n-1 > 500000

(n − 1) log 3 > log 500000

Example

n > 12.94

Therefore:

n = 13

Example:

In the earlier sequence, 400, 200, 100, 50 ... which is the first term that is less than 1?

400 × 0.5(n-1] < 1

0.5(n-1) < 0.0025

(n-1) log 0.5 < log 0.0025

Example

Therefore:

n > 9, or n = 10

Note: The inequality sign changed because we divided by a negative (log 0.5 < 0)

Sum of Geometric series

The sum of the terms can be written in two ways.

Sum

where a = first term, r = common ratio and r ≠ 1.

Sum

(use this formula when r < 1).

Example:

Evaluate,

Example

(Note: there are 9 terms.)

The first term is when n = 2

(i.e 2.362 = 5.5696)

Using the formula for the sum of a geometric progression gives:

Example

which is approximately 9300 (to 3 s.f.).

Convergence

The sum of an infinite series exists if:

-1 < r < 1 or | r | < 1

This is because each successive term is getting smaller and so the series will tend towards a certain limit. This limit is found using the second of our two formulae:

Sum

If | r | < 1 then as n → ∞, rn → 0

and so:

Example

Example:

the series 1/3 + (1/3)2 + (1/3)3 + (1/3)4 + ... converges and its sum is 1 as n approaches ∞.

(A sequence such as n3 has the first 6 terms as 1 + 8 + 27 + 64 + 125 + 216. As n approaches infinity, the sum also increases. Therefore, it is not convergent. This series is divergent.

Every AP has a sum that approaches infinity as n increases, so every AP is divergent.)

Example

Find 1 - 1/2 + 1/4 - 1/8 + ...

1 - 1/2 + 1/4 - 1/8 + ... = 1 + (-1/2) + (-1/2)2 + (-1/2)3 + ...

This is a geometric progression where r = -½, so | r | < 1.

Therefore this series converges to:

Example

Two final pieces of information that may be useful:

Arithmetic mean

The arithmetic mean of two numbers m and n is given by:

Arithmetic mean = ½(m+n)

This is the way of finding a missing term in between two known terms.

Example:

The 4th term of an AP is 14, the 6th term is 22. The 5th term will be the Arithmetic Mean of these two values.

i.e. (14 + 22)/2 = 18 (here d = 4 and a = 2).

Geometric mean

The geometric mean of two numbers m and n is given by:

Geometric mean = √(mn)

This represents the value between two others in a GP.

Example:

The 7th term of a GP is 6, the 9th is 1.5. The 8th term is:

√(6×1.5) = √9 = 3

Here r = 0.5 and a = 384.

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