Question

The sixth term of an A.P. is equal to 2, the value of
the common difference of the A.P. which makes the
product of a1, a4 and a5 least is given by??
A) 8/5
B)5/4
C)2/3
D) none​

Answer 1

Given:

The sixth term of an A.P. is equal to 2.

To find:

The value of  the common difference of the A.P. which makes the  product of a1, a4 and a5 least.

Solution:

From the given information, we have the data as follows.

The sixth term of an A.P. is equal to 2.

T_n = a + (n - 1) d

T₆ = a + (6 - 1) d

2 = a + 5d

Consider the product of a₁, a₄ and a₅ as follows.

p = a₁ × a₄ × a₅

p = (2 - 5d) × (2 - 2d) × (2 - d)

p = 2 [4 - 16d + 17d² - 5d³]

Let, the sum of terms be,

S = 4 - 16d + 17d² - 5d³

derivative of the above equation is,

S' = -15d² + 34d - 16

For S' = 0, we get,

0 = -15d² + 34d - 16

d = 2/3, 8/5

Now, again taking the derivative of S', we get,

S'' = -30d + 34

When d = 2/3, we get,

S'' = -20 + 34

S'' = 14

Therefore, the minimum value of the common difference is 2/3.