Question

Find the sum of a 9-term geometric sequence when the first term is 4 and the last term is 1,024

Answer 1

`a_1=4`

`a_2=ra_1=4r`

`a_3=ra_2=r^2a_1=4r^2`
...

`a_9=ra_8=\cdots=r^8a_1=4r^8`

So you have


`1024=4r^8\implies 256=r^8\implies r=2`

So the sum is


`4+8+16+\cdots+512+1024`

`4(1+2+4+\cdots+128+256)`

`4(2^0+2^1+2^2+\cdots+2^7+2^8)`

`4\displaystyle\sum_(i=1)^92^(i-1)`

`4*(1-2^9)/(1-2)`

`2044`