Question

# Second term of a GP is 1000 and the common ratio is r = 1n where n is a natural number. Pn is the product of n terms of this GP. P6 > P5 and P6 > P7, what is the sum of all possible values of n

A
B
C
D
##### 13
Solution

Correct option is

(B)

according to ques , common ratio is positive and one term is positive which means = all terms should be positive.

$p_{6}&space;=p&space;_{5}&space;*t_{6}&space;=&space;if&space;p_{6}>p_{5}&space;,&space;&space;t_{6}>&space;1$

$p_{7}&space;=p&space;_{6}&space;*t_{7}&space;=&space;if&space;p_{6}>p_{7}&space;,&space;&space;t_{7}<&space;&space;1$

$t_{6}&space;=t&space;_{2}&space;*r^{4}&space;=&space;1000&space;r^{4}$

$t_{7}&space;=t&space;_{2}&space;*r^{5}&space;=&space;1000&space;r^{5}$

$&space;1000&space;r^{4}&space;>1&space;and&space;1000&space;r^{5}&space;<&space;1&space;$

$&space;&space;1/r^{4}&space;<&space;1000&space;;and&space;;&space;1/&space;r^{5}&space;>&space;1&space;000$

1/r = n

$&space;&space;n^{4}&space;<&space;1000&space;=>&space;n<&space;&space;6&space;;and&space;;&space;&space;n^{5}&space;>&space;1000&space;=>&space;n&space;geq&space;4$

n can be 4 or 5 , sum of possible values = 9

so , B is the correct option