Math, asked by marshall6166, 11 months ago

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

Answered by abhi178
2

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

Answered by adventureisland
1

Option B: \frac{8}{5} is the common difference of this A.P

Explanation:

Given that a_{1}, a_{2}, a_{3}, \dots \dots be an A.P with a_{6}=2

We need to determine the common difference of this A.P which maximizes the product a_{1} a_{4} a_{5}

Let a represent the first term

Let d represent the common ratio.

Since, a_{6}=2, we have,

a+5 d=2 ------------(1)

Therefore, we get,

a_{1} \cdot a_{4} \cdot a_{5}=a(a+3 d)(a+4 d)

a_{1} \cdot a_{4} \cdot a_{5}=a(a+5d-2d)(a+5d-d)

Let us substitute a=2-5d from equation(1) , we get,

f(d)=(2-5 d)(2-2 d)(2-d)

Multiplying the terms, we get,

f(d)=(4-4d-10d+10d^2)(2-d)

       =(4-14d+10d^2)(2-d)  

       =8-4d-28d+14d^2+20d^2-10d^3

f(d)=-10d^3+34d^3-32d+8

Now, differentiating with respect to d, we get,

f'(d)=0

-30 d^{2}+68 d-32=0

Solving the quadratic equation, we get,

d=\frac{2}{3}, d=\frac{8}{5}

Now, differentiating with respect to d, we get,

f "(d)&lt;0

d=\frac{8}{5}

Thus, the common difference is \frac{8}{5}

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

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