Question

Sol. [Key Concept: A.P. with common difference $d$: maximise the product.]

$$a_{k}a_{n - (k - 1)} = \left( a_{1} + (k - 1)d \right)\left( a_{n} - (k - 1)d \right)$$

$$\begin{matrix} = & a_{1}a_{n} + d(k - 1)\left( a_{n} - a_{1} \right) - (k - 1)^{2}d^{2} \\ & \ = a_{1}a_{n} + d^{2}(k - 1)(n - k) \\ & \ \leq \left\{ \begin{matrix} a_{1}a_{n} + d^{2}\left( \frac{n + 1}{2} - 1 \right)\left( n - \frac{n + 1}{2} \right) = a_{1}a_{n} + \frac{(n - 1)^{2}}{4}d^{2}\ \text{~}\text{If}\text{~}n\text{~}\text{is odd}\text{~} \\ a_{1}a_{n} + \frac{d^{2}}{4}n(n - 2) \end{matrix} \right.\ \end{matrix}$$

✔ (a) $n$ odd ⇒ max $a_{1}a_{n} + \frac{d^{2(}n{- 1)}^{2}}{4}$

✘ (b) wrong case

✘ (c) wrong case

✔ (d) $n$ even ⇒ max $a_{1}a_{n} + \frac{d^{2\ }n(n - 2)}{4}$