Question

What is the first term in a geometric series with ten terms a common ratio of 0.5, and a sum of 511.5?

Answer 1

`256`

Explanation:

The general term of a geometric series is given by the formula:

`a_n = ar^(n-1)`

where `a` is the initial term and `r` the common ratio.

Note that:

`(1-r) sum_(n=1)^N a_n = sum_(n=1)^N ar^(n-1) - r sum_(n=1)^N ar^(n-1)`

`color(white)((1-r) sum_(n=1)^N a_n) = sum_(n=1)^N ar^(n-1) - sum_(n=2)^(N+1) ar^(n-1)`

`color(white)((1-r) sum_(n=1)^N a_n) = a+color(red)(cancel(color(black)(sum_(n=2)^N ar^(n-1)))) - color(red)(cancel(color(black)(sum_(n=2)^N ar^(n-1)))) - ar^N`

`color(white)((1-r) sum_(n=1)^N a_n) = a(1-r^N)`

Dividing both ends by `(1-r)`, we find:

`sum_(n=1)^N a_n = (a(1-r^N))/(1-r)`

In our example, we have `r=1/2`, `N=10` and:

`1023/2 = 511.5 = sum_(n=1)^10 a_n = (a(1-(color(blue)(1/2))^color(blue)(10)))/(1-color(blue)(1/2)) = 2a(1-1/1024)`

`=2a(1023/1024) = 1023/2*a/256`

Hence `a=256`