Which expression has a sum of [tex]$14$[/tex]?

A. [tex]$\sum_{k=1}^3 k$[/tex]
B. [tex]$\sum_{k=1}^3 k^2$[/tex]
C. [tex]$\sum_{k=1}^3 7k$[/tex]
D. [tex]$\sum_{k=1}^{14} k$[/tex]



Answer :

To find out which of the given expressions equals [tex]\( 14 \)[/tex], we need to individually calculate each sum and verify the resultant values.

Let's analyze each sum step by step:

### Expression (A): [tex]\(\sum_{k=1}^3 k\)[/tex]
This sum means we need to add up the integers from [tex]\(1\)[/tex] to [tex]\(3\)[/tex].

[tex]\[ \sum_{k=1}^3 k = 1 + 2 + 3 = 6 \][/tex]

So, [tex]\(\sum_{k=1}^3 k = 6\)[/tex].

### Expression (B): [tex]\(\sum_{k=1}^3 k^2\)[/tex]
This sum means we need to add up the squares of the integers from [tex]\(1\)[/tex] to [tex]\(3\)[/tex].

[tex]\[ \sum_{k=1}^3 k^2 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14 \][/tex]

So, [tex]\(\sum_{k=1}^3 k^2 = 14\)[/tex].

### Expression (C): [tex]\(\sum_{k=1}^3 7k\)[/tex]
This sum means we need to add up the values obtained by multiplying each integer from [tex]\(1\)[/tex] to [tex]\(3\)[/tex] by [tex]\(7\)[/tex].

[tex]\[ \sum_{k=1}^3 7k = 7 \times 1 + 7 \times 2 + 7 \times 3 = 7 + 14 + 21 = 42 \][/tex]

So, [tex]\(\sum_{k=1}^3 7k = 42\)[/tex].

### Expression (D): [tex]\(\sum_{k=1}^{14} k\)[/tex]
This sum means we need to add up the integers from [tex]\(1\)[/tex] to [tex]\(14\)[/tex].

We can use the formula for the sum of the first [tex]\(n\)[/tex] natural numbers, [tex]\(\frac{n(n+1)}{2}\)[/tex]:

[tex]\[ \sum_{k=1}^{14} k = \frac{14 \times (14 + 1)}{2} = \frac{14 \times 15}{2} = 105 \][/tex]

So, [tex]\(\sum_{k=1}^{14} k = 105\)[/tex].

After performing all the calculations, we find that the expression which sums to [tex]\( 14 \)[/tex] is (B):

[tex]\[ \boxed{\sum_{k=1}^3 k^2} \][/tex]