The sum of the 1?? and the last term of an AP of 150 terms

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Solution

Q.
The sum of the 1st and the last term of an AP of 150 terms is equal to the sum of which of the following two terms

1 $15 ᵗ ʰ a n d 135 ᵗ ʰ$

2 $20 ᵗ ʰ a n d 131 ˢ ᵗ$

3 $50 ᵗ ʰ a n d 51 ˢ ᵗ$

4 $90 ᵗ ʰ a n d 60 ᵗ ʰ$

Ans.
(2)

For an AP of 150 terms, the sum of the 1?? term $t e r m a_{1} a n d t h e l a s t t e r m a_{150} i s$ equal to the sum of any two terms that are equidistant from the first and last term.

This means $a_{1} + a_{150} = a_{i} + a_{151 - i}$

Given the AP has 150 terms, the middle terms are $a ₇ ₅ a n d a ₇ ₆,$ which will have the same sum as $a ₁ a n d a_{150}$

Since $a_{75} + a_{76} = a_{1} + a_{150},$ we need to find two terms $a_{i} a n d a_{j}$ from the given options that satisfy

$a_{i} + a_{j} = a_{75} + a_{76}$

From the options, the correct pair is (i, j) = (20, 131), which satisfies

$a_{i} + a_{j} = a_{1} + a_{150}$

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