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  • If in an A.P., Sn = qn^2 and Sm = qm^2, where Sr denotes the sum of r terms of the A.P., then Sq equals A. B. mnq C. q3 D. (m + n) q2

If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the A.P., then Sq equals
A. \frac{q^3}{2}
B. mnq
C. q^3
D. (m + n) q^2

Answers (1)

The given series is an A.P with first term ‘a’ and common difference ‘d’.

\\ S_{n}=\frac{n}{2} \left[ 2a+ \left( n-1 \right) d \right] \\\\ qn^{2}=\frac{n}{2} \left[ 2a+ \left( n-1 \right) d \right] \\\\ 2qn=2a+ \left( n-1 \right) d \\\\ 2a=2qn- \left( n-1 \right) d \\\\ S_{m}=\frac{n}{2} \left[ 2a+ \left( m-1 \right) d \right] \\\\ qm^{2}=\frac{m}{2} \left[ 2a+ \left( m-1 \right) d \right] \\\\

\\2qm=2a+ \left( m-1 \right) d \\\\ 2a=2qm- \left( m-1 \right) d \\\\ 2qm- \left( m-1 \right) d=2qn- \left( n-1 \right) d \\\\ 2q \left( n-m \right) = \left( n-m \right) d \\\\ d=2q \\\\ 2a=2qn- \left( n-1 \right) \left( 2q \right) \\\\

\\a=q \\\\ S_{q}=\frac{q}{2} \left[ 2q+ \left( q-1 \right) 2q \right] =q^{3} \\\\

Hence, correct option is c.

Posted by

infoexpert21

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