Question

In an A.P., the sixth term a_6​=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

The Correct Option is B

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: \(d = \frac{8}{5}\).