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Question
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Solution
The two-digit numbers which when divided by 4 yield 1 as remainder are 13, 17....97.
\[\therefore a = 13, d = 4, a_n = 97\]
\[ \therefore a_n = a + (n - 1)d\]
\[ \Rightarrow 97 = 13 + (n - 1)4\]
\[ \Rightarrow 84 = 4n - 4\]
\[ \Rightarrow 88 = 4n\]
\[ \Rightarrow 22 = n . . . \left( 1 \right)\]
\[\text { Also }, S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ S_{22} = \frac{22}{2}\left[ 2 \times 13 + (22 - 1) \times 4 \right] (\text { From }\left( 1 \right))\]
\[ \Rightarrow S_{22} = 11\left[ 110 \right] = 1210\]
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