Question

Find the seventh term of the G.P.:

`sqrt(3) + 1, 1, (sqrt(3) - 1)/2, .........`

Answer 1

Given G.P.: `sqrt(3) + 1, 1, (sqrt(3) - 1)/2, .........`

Here,

First term, a = `sqrt(3) + 1`

Common ratio, r = `1/(sqrt(3) + 1)`

Now, T_n = ar^{n – 1}

`\implies` T₇ = `(sqrt(3) + 1) xx (1/(sqrt(3) + 1))^(7 - 1)`

= `(sqrt(3) + 1) xx (1/(sqrt(3) + 1))^6`

= `(sqrt(3) + 1)/1 xx 1/(sqrt(3) + 1)^6`

= `1/(sqrt(3) + 1)^5`

= `1/(sqrt(3) + 1)^5 xx (sqrt(3) - 1)^5/(sqrt(3) - 1)^5`

= `(sqrt(3) - 1)^5/[(sqrt(3) + 1)(sqrt(3) - 1)]^5`

= `(sqrt(3) - 1)^5/(3 - 1)^5`

= `(sqrt(3) - 1)^5/(2)^5`

= `(sqrt(3) - 1)^5/32`