If the [tex]$n$[/tex]th partial sum of a sequence [tex]$a_n$[/tex] is given by

[tex]\[ \sum_{k=1}^n (2k + 4) \][/tex]

then what is the [tex]$n^{\text{th}}$[/tex] term of the sequence?

A. [tex]$2(1) + 4 + 2(n) + 4$[/tex]

B. [tex]$6 + 8 + 10 + 12 + \ldots + (2n + 4)$[/tex]

C. [tex]$2n + 4$[/tex]

D. [tex]$6$[/tex]



Answer :

To determine the [tex]\( n \)[/tex]th term of a sequence given the [tex]\( n \)[/tex]th partial sum, we need to find the difference between the [tex]\( n \)[/tex]th partial sum and the [tex]\( (n-1) \)[/tex]th partial sum.

Given the [tex]\( n \)[/tex]th partial sum [tex]\( S_n \)[/tex]:

[tex]\[ S_n = \sum_{k=1}^n (2k + 4) \][/tex]

We know that the sequence term [tex]\( a_n \)[/tex] is the difference between the [tex]\( n \)[/tex]th partial sum and the [tex]\( (n-1) \)[/tex]th partial sum. In formula terms:

[tex]\[ a_n = S_n - S_{n-1} \][/tex]

First, we need to express [tex]\( S_n \)[/tex] and [tex]\( S_{n-1} \)[/tex] explicitly:

The partial sum [tex]\( S_n \)[/tex] can be calculated as follows:

[tex]\[ S_n = \sum_{k=1}^n (2k + 4) \][/tex]

This can be simplified by separating the sum into two parts:

[tex]\[ S_n = \sum_{k=1}^n 2k + \sum_{k=1}^n 4 \][/tex]

The first part is the sum of the first [tex]\( n \)[/tex] terms of 2 times [tex]\( k \)[/tex]:

[tex]\[ \sum_{k=1}^n 2k = 2 \sum_{k=1}^n k = 2 \left(\frac{n(n+1)}{2}\right) = n(n+1) \][/tex]

The second part is the sum of 4 added [tex]\( n \)[/tex] times:

[tex]\[ \sum_{k=1}^n 4 = 4n \][/tex]

Therefore,

[tex]\[ S_n = n(n + 1) + 4n = n^2 + n + 4n = n^2 + 5n \][/tex]

Next, we calculate [tex]\( S_{n-1} \)[/tex]:

[tex]\[ S_{n-1} = (n-1)^2 + 5(n-1) \][/tex]

Expanding and simplifying:

[tex]\[ S_{n-1} = n^2 - 2n + 1 + 5n - 5 = n^2 + 3n - 4 \][/tex]

Now, we find [tex]\( a_n \)[/tex]:

[tex]\[ a_n = S_n - S_{n-1} \][/tex]

Substitute [tex]\( S_n \)[/tex] and [tex]\( S_{n-1} \)[/tex]:

[tex]\[ a_n = (n^2 + 5n) - (n^2 + 3n - 4) \][/tex]

Simplify the expression:

[tex]\[ a_n = n^2 + 5n - n^2 - 3n + 4 = 2n + 4 \][/tex]

Thus, the [tex]\( n \)[/tex]th term of the sequence is:

[tex]\[ 2n + 4 \][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{C. \, 2n + 4} \][/tex]