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प्रश्न
Write the common difference of an A.P. the sum of whose first n terms is
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उत्तर
Sum of the first n terms of an A.P. = \[\frac{p}{2} n^2 + Qn\]
Sum of one term of an A.P. = \[S_1\]
\[\Rightarrow \frac{p}{2} \left( 1 \right)^2 + Q\left( 1 \right)\]
\[ \Rightarrow \frac{p}{2} + Q\]
Sum of two terms of an A.P. =
\[\Rightarrow \frac{p}{2} \left( 2 \right)^2 + Q\left( 2 \right)\]
\[ \Rightarrow 2p + 2Q\]
Now, we have:
\[a_1 + a_2 = S_2 \]
\[ \Rightarrow \frac{p}{2} + Q + a_2 = 2p + 2Q\]
\[ \Rightarrow a_2 = Q + \frac{3}{2}p\]
Common difference:
\[d = a_2 - a_1 \]
\[ = \left( Q + \frac{3}{2}p \right) - \left( Q + \frac{p}{2} \right)\]
\[ = p\]
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