Question

Given two terms from a geometric sequence, identify the first term and the common ratio: a10 = 1 and a12=1/25

Answer 1

Given:

a denotes first term and r denotes the common ratio.

`a_(10)=1\colon a_(12)=(1)/(25)`
`a_n=ar^(n-1)`
`a_(10)=ar^(10-1)`
`1=ar^9\ldots.\text{ (1) }`
`a_(12)=ar^(12-1)`
`(1)/(25)=ar^(11)\ldots.(2)`

Divide the equation (2) by (1)

`((1)/(25))/(1)=(ar^(11))/(ar^9)`
`(1)/(25)=r^2`
`r=\pm(1)/(5)`
`\text{If r=}(1)/(5)`
`1=a((1)/(5))^9`
`a=1953125`
`\text{If r=-}(1)/(5)`
`1=a(-(1)/(5))^9`
`a=-1953125`
`a=-1953125\text{ ; r = -}(1)/(5)`
`a=1953125\text{ ; r = }(1)/(5)`