Question

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

1. 5
2. 9
3. 8
4. 11

Explanatory Answer

Common ratio is positive, and one of the terms is positive => All terms are positive

P₆ = P₅ * t₆ => If P₆ > P₅, t₆ > 1

P₇ = P₆ * t₇ => If P₆ > P₇, t₇ < 1

t₆ = t₂ * r⁴ = 1000r⁴;

t₇ = t₂ * r⁵ = 1000r⁵

1000r⁴ > 1 and 1000r⁵ < 1

< 1000 and > 1000

= n

n⁴ < 1000 and n⁵ > 1000, where n is a natural number

n⁴ < 1000 => n < 6

n⁵ > 1000 => n ≥ 4

n could be 4 or 5. Sum of possible values = 9

Answer Choice (B)