Arithmetic and Geometric Progressions

When learning about sequences and series, arithmetic and geometric progressions are two of the most important types of sequences to learn about. Carry on reading for a simple explanation of each type followed by some worked examples.

Arithmetic Progressions

Arithmetic Progression – a sequence where each following term is found by adding a constant.

This basically means to get the next term in the sequence you keep adding the same value to the previous term each time.

For example:

3, 5, 7, 9, 11…

In this sequence the constant is 2. By adding 2 to the previous term we get the next term.

For arithmetic progressions we denote the first term a and the common difference (the constant value) d.

So in the sequence above a = 3 and d = 2.

To find the n^th term in an arithmetic progression we can use the following equation:

U_n represents the n^th term. So U₁ would be the first term where n=1 and U₂ would be the second term where n=2 etc. This is the same for all equations in this post.

To find the sum of the first n terms we can use this equation:

S_n represents the sum of the first n terms. So U₂ would be the sum of the first two terms and U₃ would be the sum of the first three terms etc. This is the same for all equations in this post.

Geometric Progressions

Geometric Progression – a sequence where each following term is found by multiplying the previous term by a common ratio.

This basically means to get the next term in the sequence you keep multiplying the previous term by the same value.

For example:

3, 9, 27, 81, 243…

In this sequence the common ratio is 3. By multiplying the previous term by 3 we get the next term.

As with arithmetic progressions we denote the first term a. We denote the common ratio (the constant we are multiplying by) r.

So in the sequence above a = 3 and r = 3.

To find the n^th term in a geometric progression we can use the following equation:

To find the sum of the first n terms we can use the following equation when r < 1:

And this equation for when r > 1:

r can be found by using consecutive term:

So in the sequence above we could find r by dividing the second term by the first term: 9 ÷ 3 = 3, or the third term by the second term: 27 ÷ 9 = 3 etc.

Arithmetic Progressions Examples

Example 1

Find the sum of the first 10 terms in the following sequence:

4, 10, 16, 22…

Example 2

Find the number of terms in the following sequence:

7, 9, 11 … 65, 67, 69

Example 3

The n^th term of a sequence is given by 1 + 4n. Find a, d, and the 12^th term of the sequence.

Geometric Progressions Examples

Example 1

Find the sum of the first 6 terms in the following sequence:

5, 7.5, 11.25…

Example 2

Find a and r when U₄ = 48 and U₆ = 192.

Example 3

Find the 8^th term in the following sequence:

3, 12, 48…

Please leave any questions you have in the comments below.

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