The sum of the first $10$ terms of the AP is $100$ and the sum of next $10$ terms is $300$ . Find the first term and the common difference.

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Solution

Given sum of first $10$ terms of A.P $= 100$
next $10$ terms $= 300$ , common difference $= 100$
sum of first $20$ terms $= 400$
$\frac{n}{2} [2 a + (n - 1) d] = 100$
$3 (2 a + 9 d) = 100$
$2 a + 9 d = 20.... (1)$
$\frac{20}{2} (2 a + 19 d) = 400$
$2 a + 19 d = 40.... (2)$
from (1) and (2) $a = 1; d = 2$

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