The sum of the terms of an infinitely decreasing GP is equal to the greatest value of the function f ( x ) = x 3 + 3 x − 9 on the interval [-4,3] and the difference between the first and second terms

Question

The sum of the terms of an infinitely decreasing GP is equal to the greatest value of the function f ( x ) = x 3 + 3 x − 9 on the interval [-4,3] and the difference between the first and second terms is f ′ ( 0 ) . Then find the value of 27 r where is common ratio.

TestHub · Accepted Answer

f is increasing, so its greatest value is f ( 3 ) = 27 . Let the GP be a , a r , a r 2 , … with − 1 < r < 1 . 1 − r a ​ = 27 And a − a r = 3 ⇒ r = 3 4 ​ or r = 3 2 ​ . But since − 1 < r < 1 , we have r = 3 2 ​ . ⇒ 27 r = 18

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