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 problem involves an arithmetic progression (A.P.) where the sixth term \(a_6 = 2\). We need to find the common difference \(d\) that maximizes the product of three terms: \(a_1 \times a_4 \times a_5\).

Let's analyze the problem step-by-step: 

1. In an arithmetic progression, the nth term can be given by the formula: \(a_n = a_1 + (n-1) \cdot d\).
2. Thus, the sixth term \(a_6\) can be written as: \(a_6 = a_1 + 5d = 2\).
3. From the above, we get: \(a_1 = 2 - 5d\).
4. Now we know:
   • \(a_1 = 2 - 5d\),
   • \(a_4 = a_1 + 3d = (2 - 5d) + 3d = 2 - 2d\),
   • \(a_5 = a_1 + 4d = (2 - 5d) + 4d = 2 - d\).
5. The product \(a_1 \times a_4 \times a_5\) can be written as: \((2 - 5d)(2 - 2d)(2 - d)\).
6. To maximize this product, we consider the roots of these linear expressions, which hints that \((2 - 5d)(2 - 2d)(2 - d)\) should be zero or near zero for one of its factors. To find the critical points, we look at the value of \(d\) that could make two of the factors equalized.
7. A trial and error or mathematical optimization technique can help find that setting \(d\) to an appropriate value within given options maximizes the product.

After testing the values, choosing \(d = \frac{8}{5}\), you find that it results in equalizing terms and maximizing the product due to symmetry. Therefore, the common difference \(d\) is \(\frac{8}{5}\).

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: \(d = \frac{8}{5}\).