Question

If the sum of the first 12 of a geometric series is 8190 and the common ratio is 2. Find the first term and the 20th term.

Answer 1

`a_1=2`

`a_(20)=1048576`

Explanation:

Geometric Series

The sum of the first n terms of a geometric series whose first term is a1 and a common ratio r is

`\displaystyle S_n=a_1\cdot (r^n-1)/(r-1)`

The problem provides the following data

`S_(12)=8190`

`r=2`

`n=12`

Replacing the values in the formula:

`\displaystyle 8190=a_1\cdot (2^(12)-1)/(2-1)`

Operating

`\displaystyle 8190=a_1\cdot 4095`

Solving for a1

`a_1=8190/4095=2`

`\boxed{a_1=2}`

The first term is 2

The general term of the series is

`a_n=a_1.r^(n-1)`

Compute the 20th term

`a_(20)=2\cdot 2^(20-1)`

`\boxed{a_(20)=1048576}`