If two AP's have the same first term and the difference between their common difference is 2, then the difference between the sum of their first 10 terms is

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100

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Solution

The correct option is C 90
Let $a$ be the first terms of both the APs and $d_{1}$ and $d_{2}$ be their common difference respectively.
Given, $d_{1} - d_{2} = 2$
Also, Sum to $n$ terms of an AP $= \frac{n}{2} (2 a + (n - 1) d)$
So, difference of sum of their first $10$ terms , $= S_{1} - S_{2} = \frac{10}{2} (2 a + (10 - 1) d_{1}) - \frac{10}{2} (2 a + (10 - 1) d_{2})$

$= 5 (2 a + 9 d_{1} - 2 a - 9 d_{2}) = 45 (d_{1} - d_{2}) = 45 \times 2 = 90$

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