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Last updated date: 04th Dec 2023
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# 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. $Q$

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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.

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.