To determine which of the choices is equivalent to [tex]\(\sum_{k=0}^5 a_k\)[/tex], let's break down the summation symbol [tex]\(\sum_{k=0}^5 a_k\)[/tex]:
The summation symbol [tex]\(\sum_{k=0}^5 a_k\)[/tex] tells us to sum the terms [tex]\(a_k\)[/tex] starting from [tex]\(k=0\)[/tex] and continuing up to [tex]\(k=5\)[/tex]. This means:
[tex]\[
\sum_{k=0}^5 a_k = a_0 + a_1 + a_2 + a_3 + a_4 + a_5
\][/tex]
Now, let's analyze each choice:
A) [tex]\(a_0 + a_1 + a_2 + \ldots\)[/tex]
This suggests an ongoing sum that starts at [tex]\(a_0\)[/tex] but does not specify an end. It implies the sum continues indefinitely, which is not what [tex]\(\sum_{k=0}^5 a_k\)[/tex] represents.
B) [tex]\(a_1 + a_2 + a_3 + \ldots + a_{\infty}\)[/tex]
This starts at [tex]\(a_1\)[/tex] and continues indefinitely, essentially implying an infinite sum starting from [tex]\(a_1\)[/tex], which again is not what [tex]\(\sum_{k=0}^5 a_k\)[/tex] represents.
C) [tex]\(a_0 + a_1 + a_2 + a_3 + a_4 + a_5\)[/tex]
This matches exactly what we expanded from [tex]\(\sum_{k=0}^5 a_k\)[/tex]. It includes each term from [tex]\(a_0\)[/tex] to [tex]\(a_5\)[/tex].
D) [tex]\(a_1 + a_2 + a_3 + a_4 + a_5\)[/tex]
This starts from [tex]\(a_1\)[/tex] and goes up to [tex]\(a_5\)[/tex], thereby missing the [tex]\(a_0\)[/tex] term. Hence, it doesn't match [tex]\(\sum_{k=0}^5 a_k\)[/tex].
Therefore, the correct choice that matches [tex]\(\sum_{k=0}^5 a_k\)[/tex] is:
C) [tex]\(a_0 + a_1 + a_2 + a_3 + a_4 + a_5\)[/tex]
