Let a1, a2, a3,………. be an A.P. with
Question:

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots . .$ be an A. P. with $\mathrm{a}_{6}=2$. Then the common difference of this A. P., which maximises the produce $\mathrm{a}_{1} \mathrm{a}_{4} \mathrm{a}_{5}$, is :

  1. $\frac{6}{5}$

  2. $\frac{8}{5}$

  3. $\frac{2}{3}$

  4. $\frac{3}{2}$


Correct Option: , 2

Solution:

Let a is first term and $\mathrm{d}$ is common difference then, a $+5 \mathrm{~d}=2$ (given) …(1)

$f(d)=(2-5 d)(2-2 d)(2-d)$

$\mathrm{f}^{\prime}(\mathrm{d})=0 \quad \Rightarrow \mathrm{d}=\frac{2}{3}, \frac{8}{5}$

$\mathrm{f}^{\prime \prime}(\mathrm{d})<0$ at $\mathrm{d}=8 / 5$

$\Rightarrow \mathrm{d}=\frac{8}{5}$

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