Let a₁, a₂, a₃, …… be and A.P with a₆ = 2. Then the common difference of this A.P., which maximizes the product a₁a₄a₅ is (A) 3/2
(B) 8/5
(C) 2/3
(D) 6/5
Answers
answer : option (B) 8/5
given, Let a₁, a₂, a₃, …… be and A.P with a₆ = 2.
so, a + 5d = 2 .......(1)
let a is the first term and d is the common difference.
a₁a₄a₅ = a(a + 3d)(a + 4d)
= (2 - 5d)(2 - 5d + 3d)(2 - 5d + 4d) [ from equation (1). ]
= (2 - 5d)(2 - 2d)(2 - d)
so, f(d) = a₁a₄a₅ = (2 - 5d)(2 - 2d)(2 - d)
differentiating with respect to d,
f'(d) = -5(2 - 2d)(2 - d) -2(2 - 5d)(2 - d) -(2 - 5d)(2 - 2d)
= -2(15d² - 34d + 16)
for f'(d) = 0 ⇒d = 2/3 , 8/5
now differentiating once again,
f"(d) = -2(30d - 34)
at d = 8/5 , f"(d) < 0
hence f(d) is maximum at d = 8/5
Option B: is the common difference of this A.P
Explanation:
Given that be an A.P with
We need to determine the common difference of this A.P which maximizes the product
Let a represent the first term
Let d represent the common ratio.
Since, , we have,
------------(1)
Therefore, we get,
Let us substitute from equation(1) , we get,
Multiplying the terms, we get,
Now, differentiating with respect to d, we get,
Solving the quadratic equation, we get,
Now, differentiating with respect to d, we get,
Thus, the common difference is
Therefore, Option B is the correct answer.
Learn more:
(1) Find common difference of Ap whose first term is 5 and the sum of its four terms is half the sum of next four terms.
brainly.in/question/14245089
(2) Common difference of ap root 7,root 28,root63
brainly.in/question/2389248