Answer 1

101 + 99 + 97 + ... + 47
Here, the series is an A.P. where we have the following:

\[a = 101\]

\[d = \left( 99 - 101 \right) = - 2\]

\[ a_n = 47\]

\[ \Rightarrow 101 + (n - 1)( - 2) = 47\]

\[ \Rightarrow 101 - 2n + 2 = 47\]

\[ \Rightarrow 2n - 2 = 54\]

\[ \Rightarrow 2n = 56\]

\[ \Rightarrow n = 28\]

\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]

\[ \Rightarrow S_{28} = \frac{28}{2}\left[ 2 \times 101 + \left( 28 - 1 \right) \times ( - 2) \right]\]

\[ = \frac{28}{2}\left[ 2 \times 101 + 27 \times ( - 2) \right] \]

\[ = 2072\]