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Question
How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?
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Solution
\[\text { Given: } \]
\[\text{ An } \hspace{0.167em} A . P .\text { with a = - 6 and d }= - \frac{11}{2} - \left( - 6 \right) = \frac{1}{2}\]
\[ S_n = - 25\]
\[ \therefore - 25 = \frac{n}{2}\left[ 2 \times \left( - 6 \right) + \left( n - 1 \right)\frac{1}{2} \right]\]
\[ \Rightarrow - 25 = \frac{n}{2}\left[ - 12 + \frac{n}{2} - \frac{1}{2} \right]\]
\[ \Rightarrow - 50 = n\left[ \frac{n}{2} - \frac{25}{2} \right]\]
\[ \Rightarrow - 100 = n\left( n - 25 \right)\]
\[ \Rightarrow n^2 - 25n + 100 = 0\]
\[ \Rightarrow \left( n - 20 \right)\left( n - 5 \right) = 0\]
\[ \Rightarrow n = 20 \text { or } n = 5\]
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