Question

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 1

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.