June 26, 2019

A geometric sequence or geometric progression is a sequence of numbers such that the ratios between successive terms is a constant **r**, called the common ratio.

3, 6, 12, 24, 48, ...

6/3 = 2.

12/6 = 2.

24/12 = 2.

48/24 = 2.

r = 2.

### Nth Term of a Geometric Sequence

1If the 1st term is known.

**a _{n} = a_{1} · r^{n-1}**

3, 6, 12, 24, 48, ..

a_{n} = 3 · 2^{n-1} = 3 · 2^{n} · 2^{-1} = (3/2)· 2^{n}

2If the value that occupies any other term of the sequence is known.

**a _{n} = a_{k} · r^{n-k}**

a_{4}= 24, k=4 and r=2.

a_{n} = a_{4} · r^{n-4}

a_{n} = 24· 2^{n-4}= (24/16)· 2^{n }= (3/2) · 2^{n}

### Geometric Series

A geometric series is the sum of a geometric sequence.

Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

### Infinite Geometric Series

If −1 < r < 1 the infinite geometric series converges to a specific value:

Calculate the sum of the terms of the sequence:

### Product

Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

Find the ninth term of the geometric sequence 3,6,12,..