Question

In an A.P., the sixth term a6​=2. If the product a1​a4​a5 is the greatest, then the common difference of the A.P. is equal to:

• A. $\frac{3}{2}$
• B. $\frac{8}{5}$
• C. $\frac{2}{3}$
• D. $\frac{5}{8}$

Answer 1

Answer: (B)

$\frac{8}{5}$

Answer 2

The sixth term of an A.P. can be expressed as:
$a_6 = a + 5d = 2$
were $a$ is the first term and $d$ is the common difference. Therefore, we have:
a = 2 - 5d

The product $a_1 a_4 a_5$ can be expressed as:
$a_1 a_4 a_5 = a(a + 3d)(a + 4d)$
Substituting $a = 2 - 5d$ into this expression, we get:
$a_1 a_4 a_5 = (2 - 5d)(2 - 2d)(2 - d)$

To find the maximum value of this product, we can analyze the behavior of the function:
$f(d) = (2 - 5d)(2 - 2d)(2 - d)$

After taking the derivative and setting it to zero, the solution in the image calculates critical points and finds that $d = \frac{8}{5}$ maximizes the product.

So, the correct option is: $d = \frac{8}{5}$.