Question

Sum of first 12 terms of a gp is equal to the sum of the first 14 terms in the same gp. Sum of the first 17 terms is 92, what is the third term in the gp?

Answer 1

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Answer 2

Third term of the GP is 92.

• Sum of first 12 terms of the GP is equal to first 14 terms of the GP.

$S_12$ = $S_14$

$S_12$ = $S_12 + t_13 + t_14$

$t_13 + t_14$ = 0      (Equation 1)

• Let the $13^{th}$ term of the GP be x.

• And the Common ratio be r. Therefore $14^{th}$ term will be xr.

x + xr = 0    (From equation 1)

x ( 1 + r ) = 0

r = -1

• As the common ratio is -1.

• Now sum of first 17 terms of GP is 92.

$S_{17}$ = 92

• The sum of n terms of Gp when r <1 is

$S_n$ = $\frac{a(1-r^n)}{1-r}$

92 = $\frac{a(1-{(-1)}^{17})}{1-(-1)}$

a(first term) = 92

• Now the first term of the GP is 92.

• Now we have to find the 3rd term.

• nth term of GP is given by

$t_n$ = $ar^{n-1}$

$3^{rd}$ = $92.{(-1)}^{16}$ = 92