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If \(S_n = nP + \frac{n(n-1)}{2}Q\), where Sn is the sum of n terms of an arithmetic progression, then its common difference is
1. P + Q
2. 2P + 3Q
3. 2Q
4. Q

1 Answer

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Best answer
Correct Answer - Option 4 : Q

Concept:

Sum of n AP terms = \({{\rm{S}}_{\rm{n}}} = \frac{{\rm{n}}}{2}\left[ {2{\rm{a}} + \left( {{\rm{n}} - 1} \right){\rm{d}}} \right]\) 

Where a is the first term of AP and d is the difference between two consecutive terms of AP.

Calculation:

Given that,

\(S_n = nP + \frac{n(n-1)}{2}Q\)

\(\Rightarrow \ S_n = \frac{n}{2}[2P + (n-1)Q]\) ----(1)

We know that the sum of the nth term of AP is 

\({{\rm{S}}_{\rm{n}}} = \frac{{\rm{n}}}{2}\left[ {2{\rm{a}} + \left( {{\rm{n}} - 1} \right){\rm{d}}} \right]\)    ----(2)

From equation (1) and (2)

a = P, d = Q 

Hence, the first term of the A.P is P and the common difference is Q.

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