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Question
If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
Options
\[\frac{1}{2} p^3\]
mn p
P3
(m + n) p2
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Solution
p3
Given:
\[S_n = n^2 p\]
\[ \Rightarrow \frac{n}{2}\left\{ 2a + \left( n - 1 \right)d \right\} = n^2 p\]
\[ \Rightarrow 2a + \left( n - 1 \right)d = 2np\]
\[ \Rightarrow 2a = 2np - \left( n - 1 \right)d . . . . . \left( 1 \right)\]
\[ S_m = m^2 p\]
\[ \Rightarrow \frac{m}{2}\left\{ 2a + \left( m - 1 \right)d \right\} = m^2 p\]
\[ \Rightarrow 2a + \left( m - 1 \right)d = 2mp\]
\[ \Rightarrow 2a = 2mp - \left( m - 1 \right)d . . . . . \left( 2 \right)\]
From
\[\left( 1 \right) \text { and } \left( 2 \right)\] , we have:
\[2np - \left( n - 1 \right)d = 2mp - \left( m - 1 \right)d\]
\[ \Rightarrow 2p\left( n - m \right) = d\left( n - 1 - m + 1 \right)\]
\[ \Rightarrow 2p = d\]
Substituting d = 2p in equation \[\left( 1 \right)\], we get:
a = p
Sum of p terms of the A.P. is given by:
\[\frac{p}{2}\left\{ 2a + \left( p - 1 \right)d \right\}\]
\[ = \frac{p}{2}\left\{ 2p + \left( p - 1 \right)2p \right\} \]
\[ = p^3\]
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