Question

For any integer , the sum \[_{k = 1}^{k = n}{S_n}k(k + 2)\] is equal to
1) \[\dfrac{{n(n + 1)(n + 2)}}{2}\]
2) \[\dfrac{{n(n + 1)(2n + 1)}}{6}\]
3) \[\dfrac{{n(n + 1)(2n + 7)}}{6}\]
4) \[\dfrac{{n(n + 1)(2n + 9)}}{6}\]

Answer 1

Hint: This is a basic question of sequences and series. First simplify the expression into standard form and then apply the appropriate formulas. The formulas involved in this question are:
the formula for the sum of the n consecutive integers, \[\dfrac{{n(n + 1)}}{2}\] and the formula for the sum of the squares of n positive integers, \[\dfrac{{n(n + 1)(2n + 1)}}{6}\].

Complete step-by-step answer:
 Let’s begin the question by writing out the expression we have i.e.,
\[ \Rightarrow \sum\limits_{k = 1}^n {k(k + 2)} \]
Now, let’s open the brackets for the expression as shown below
\[ \Rightarrow \sum\limits_{k = 1}^n {{k^2} + 2k} \]
Now, separating the two parameters under the limit keeping everything else constant we get,
\[ \Rightarrow \sum\limits_{k = 1}^n {{k^2}} + \sum\limits_{k = 1}^n {2k} \]
Now, taking the constant term out from the second term while keeping everything else constant, we \[ \Rightarrow \dfrac{{n(n + 1)(2n + 1)}}{6} + \dfrac{{n(n + 1)}}{2} \times 2\]\[ \Rightarrow \sum\limits_{k = 1}^n {{k^2}} + 2\sum\limits_{k = 1}^n k \]
Clearly, the first part of the expression depicts the sum of the squares of n consecutive natural numbers while the second part depicts the sum of n consecutive natural numbers. We know the expression for the sum of n natural numbers as \[\dfrac{{n(n + 1)}}{2}\] and sum of squares of n natural numbers as \[\dfrac{{n(n + 1)(2n + 1)}}{6}\]. So, we get,
After replacing their values, we find \[n(n + 1)\]is common in both the expressions, therefore taking it out while keeping everything else constant, we get,
\[ \Rightarrow n(n + 1)\left[ {\dfrac{{(2n + 1)}}{6} + 1} \right]\]
Now, solving the bracket we get,
\[ \Rightarrow n(n + 1)\left[ {\dfrac{{2n + 7}}{6}} \right]\]
And finally, we arrive at the final answer
\[ \Rightarrow \dfrac{{n(n + 1)(2n + 7)}}{6}\]
So, the correct answer is “Option 3”.

Note: In mathematics, a series is the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence.