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Example 5 - If sum of n terms of AP is nP + 1/2n(n-1)Q - Arithmetic Progression (AP): Calculation based/Proofs

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
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Example 5, If the sum of n terms of an A.P. is nP + 1/2n(n –1)Q , where P and Q are constants, find the common difference. Let a1, a2, … an be the given A.P Given, Sum of n terms = nP + 1/2 n (n – 1) Q Sn = nP + 1/2 n (n – 1) Q Putting n = 1 in (1) S1 = 1 × P + 1/2 × 1 × (1 – 1)Q S1 = P + 1/2(0) S1 = P But sum of first 1 terms will be the first term ∴ a1 = S1 = P Sn = nP + 1/2 n (n – 1) Q …(1) Putting n = 2 in (1) S2 = 2P + 1/2 × 2(2 – 1)Q S2 = 2P + 1/2 × 2 × 1 × Q S2 = 2P + Q Sum of first two terms = First term + Second term S2 = a1 + a2 S2 – a1 = a2 a2 = S2 – a1 Putting a1 = P , S2 = 2P + Q a2 = 2P + Q – P = 2P – P + Q = P + Q Thus, a2 = P + Q Common difference (d) = a2 – a1 = (P + Q) – P = Q Hence, common difference of the given A.P. is Q

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