Explanation
Concept:
GP is represented as a,ar,ar2,ar3,……………,arn, where a is the first term, r is the common ratio.
The sum of infinite terms of a decreasing GP = 1−ra
Calculation:
f(x)=x3+3x−9
f′(x)=3x2+3=3(x2+1)≥3 for all x∈R
f(x) is strictly increasing function for all x∈R
so the greatest value of f(x) in the interval [−2,3] is at x=3 So, let's find f(3)
⇒f(3)=27
According to question we know that the sum of infinite terms of a decreasing GP is equal to the greatest value of the function f(x)
⇒1−ra=27
⇒a=27×(1−r)
According to question, the difference between the first two terms is f(0)
⇒a−ar=f(0)=3
⇒a(1−r)=3
So, using equation (1) and (2) we get,
27×(1−r)×(1−r)=3
(1−r)2=1/9
(1−r)=±1/3
⇒r=32 or 34
GP is decreasing series so common ratio r=32
