# If $ {S_n} = nP + \dfrac{{n(n - 1)}}{2}Q $ , where $ {S_n} $ denotes the sum of the first $ n $ terms of an A.P., then the common difference is?

A. $ P + Q $

B. $ 2P + 3Q $

C. $ 2Q $

D. $ Q $

Answer

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**Hint**: The term a.p. given in the question refers to the term arithmetic progression. The numbers in an arithmetic progression differ by a common number $ d $ , called as the common difference, this is the term which the question asks. The first term is denoted by $ a $ ,and the sum of first $ n $ terms of an arithmetic progression is given by

$ {S_n} = \dfrac{n}{2}(2a + (n - 1)d) $

The term $ a $ is the first term of the progression and the term $ d $ is the common difference of the progression.

We will use this formula to solve our given question.

**Complete step-by-step answer**:

The question asks us to find the common difference $ d $ of the arithmetic progression, the formula for the first $ n $ terms of the arithmetic progression is given by,

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

This can be compared with the generic formula to find the value of common difference, the generic formula is ,

$ {S_n} = \dfrac{n}{2}(2a + (n - 1)d) $

Let the first term be $ a $ then the sum of $ n $ terms will be,

$ \Rightarrow na + \dfrac{{n(n - 1)d}}{2} $

Upon comparing this with the given formula we can write,

$ \Rightarrow na + \dfrac{{n(n - 1)d}}{2} = nP + \dfrac{{n(n - 1)}}{2}Q $

The term $ a $ is $ P $ and the term $ d $ is $ Q $ ,

Thus we can say the term $ d $ is the $ Q $ , which is the option D.

**So, the correct answer is “Option D”.**

**Note**: The sum of the arithmetic progression can also be expressed in the terms of the first and the last terms of the progression.

\[{S_n} = \dfrac{n}{2}(a + l)\]

Where the term $ a $ is the first term in the progression and the term $ l $ is the last term in the progression. The term $ n $ is the number of terms in the arithmetic progression.

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