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Question
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 ______.
Options
`q^3/2`
mnq
q3
(m + n)q2
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Solution
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 q3.
Explanation:
The given series is A.P. whose first term is a and common difference is d
∴ Sn = `n/2[2a + (n - 1)d]` = qn2
⇒ 2a + (n – 1)d = 2qn ....(i)
Sm = `m/2 [2a + (m - 1)d]` = qm2
⇒ 2a + (m – 1)d = 2qm .....(ii)
Solving equation (i) and equation (ii) we get
2a + (m – 1)d = 2qm
2a + (n – 1)d = 2qn
(–) (–) (–)
(m – n)d = 2qm – 2qn
(m – n)d = 2q(m – n)
∴ d = 2q
Putting the value of d in equation (ii) we get
2a + (m – 1) · 2q = 2qm
⇒ 2a = 2qm – (m –1)2q
⇒ 2a = 2q(m – m + 1)
⇒ 2a = 2q
⇒ a = q
∴ Sq = `q/2 [2a + (q - 1)d]`
= `q/2[2q + (q - 1)2q]`
= `q/2[2q + 2q^2 - 2q]`
= `q/2 xx 2q^2`
= q3
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