Question

If 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 a₁ a₄ a₅ the greatest is (the i^th term is denoted by a)

• A. $\frac{8}{5}$
• B. 3
• C. 2
• D. $\frac{4}{5}$

Answer 1

Answer: (A)

$\frac{8}{5}$

Answer 2

a₁ + 5x = 2

Ž a₁ = 2 – 5x

now, a₁. a₄. a₅ = (2 – 5x)(2 – 5x + 3x) (2 –5x + 4x)

= (2 – 5x) (2 – 2x) (2 – x)

= 8 – 12x + 4x² – 20x + 30x² – 10x³

p = 8 – 32x + 34x² – 10x³

Ž$\frac{dp}{dx}$ = – 32 + 68x – 30x² = 0

x = $\frac{2}{3},\frac{8}{5}$

$\frac{d^{2}p}{dx^{2}}$ = 68 – 60x

for x = $\frac{8}{5}$, $\frac{d^{2}p}{dx^{2}}$ < 0

common difference = $\frac{8}{5}$.