Complete the AP: _ , 3, _ , -5.

7, -1

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-7, 1

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-7, -1

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7, 1

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Solution

The correct option is A 7, -1
Let the common difference for this AP be $d .$

Given that the $2^{nd}$ term in this AP is 3, and the $4^{th}$ term is -5.
i.e., $a_{2} = 3$ and $a_{4} = - 5$

Since $a_{3} = a_{2} + d$ and $a_{4} = a_{3} + d,$ we have
$a_{4} = a_{3} + d = (a_{2} + d) + d = a_{2} + 2 d .$
$⟹ - 5 = 3 + 2 d$
$⟹ d = - 4$

$a_{2} = a_{1} + d$
$⟹ a_{1} = a_{2} - d = 3 - (- 4) = 7$
Thus, the $1^{st}$ term of the AP is 7.

$a_{4} = a_{3} + d$
$⟹ a_{3} = a_{4} - d = - 5 - (- 4) = - 1$
Thus, the $3^{rd}$ term of the AP is -1.

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