Q.146. Arithmetical Progression

Question 146.
If the sum of  'n' terms of an AP is (pn+qn^2), where p and q are constants, find the common difference.

Answer 146.
Sn = pn + qn^2
=> S(n-1) = p(n-1) + q(n-1)^2
=> tn
= Sn - S(n - 1)
= pn + qn^2 - p(n - 1) - q(n - 1)^2
= p + 2qn - q

Common difference
= tn - t(n-1)
= (p - q + 2qn) - [p - q + 2q(n - 1)]
= 2q.

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Verification:
Putting n = 1 in tn,
t1 = p + q
Sn
= (n/2) [2t1 + (n - 1) * (common diff.)]
= (n/2) [2p + 2q + (n - 1) * (2q)]
= (n/2) [2p + 2qn]
= pn + qn^2.

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