The first term of an arithmetic progression $a_{1}, a_{2}, a_{3}$ _____ is equal to unity then the value of the common difference of the progression if $a_{1} a_{3} + a_{2} a_{3}$ is minimum, is

- 5 / 4

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- 5 / 2

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4

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None of these

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Solution

The correct option is A $- 5 / 4$
According to the problem, $a_{1} = 1$ and let $d$ be the common difference of the A.P.
Then $a_{2} = 1 + d$ and $a_{3} = 1 + 2 d$ .
Now, $a_{1} a_{3} + a_{2} a_{3}$
$= (a_{1} + a_{2}) a_{3}$
$= (2 + d) (1 + 2 d) = f (d)$ [ Let]
Then $f (d) = 2 + 5 d + 2 d^{2}$
This gives,
$f^{'} (d) = 5 + 4 d$ and $f^{''} (d) = 4$ .
Now for maximum or minimum value of $f (d)$ we must have $f (d) = 0$
or, $d = - \frac{5}{4}$ .
Since $f^{''} (d) > 0 \forall d$ .
So $d = - \frac{5}{4}$ we get minimum value of $f (d)$ .

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The first term of an arithmetic progression a1,a2,a3_____ is equal to unity then the value of the common difference of the progression if a1a3+a2a3 is minimum, is

The first term of an arithmetic progression $a_{1}, a_{2}, a_{3}$ _____ is equal to unity then the value of the common difference of the progression if $a_{1} a_{3} + a_{2} a_{3}$ is minimum, is