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]
