Question

Determine the number of terms of a G.P. if the first term = 3, the n^th term = 96 and the sum of n terms = 189.

Answer 1

a = 3

a_n = 96

S_n = 189

a_n = ar^{n − 1}

96 = 3.r^{n − 1}

`96/3` = r^{n − 1}

32 = r^{n − 1}

`S_n = (a(r^n - 1))/((r - 1))`

`189 = (3(r^n - 1))/(r - 1)`

`189/3 = ((r^n - 1))/((r - 1))`

`189/3 = ((r^(n - 1) - 1/r)/(1 - 1/r))`

63 = `((32 - 1/r))/((1 - 1/r))`

63 = `(((32r - 1)/r)/((r - 1)/4))`

63 = `(32r - 1)/(r - 1)`

63r − 63 = 32r − 1

32 = r^{n − 1}

(2)⁵ = (2)^{n − 1}

5 = n − 1

5 + 1 = n

n = 6