(a) Find the value of [tex]\( T \)[/tex] when [tex]\( n = 15 \)[/tex].
We start by plugging [tex]\( n = 15 \)[/tex] into the given formula for [tex]\( T \)[/tex]:
[tex]\[
T = \frac{n(n+1)}{2}
\][/tex]
Substituting [tex]\( n = 15 \)[/tex] into the equation:
[tex]\[
T = \frac{15 \cdot (15 + 1)}{2}
\][/tex]
Next, simplify inside the parentheses:
[tex]\[
T = \frac{15 \cdot 16}{2}
\][/tex]
Now, multiply 15 and 16:
[tex]\[
T = \frac{240}{2}
\][/tex]
Finally, divide by 2:
[tex]\[
T = 120.0
\][/tex]
Write your answer in the box below:
[tex]\[
\boxed{120.0}
\][/tex]
(b) Show a check of your answer.
To check our result, we will sum all integers from 1 to [tex]\( n \)[/tex] where [tex]\( n = 15 \)[/tex].
[tex]\[
\text{Sum} = 1 + 2 + 3 + \dots + 15
\][/tex]
We can list and add these numbers directly:
[tex]\[
\text{Sum} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15
\][/tex]
Performing the addition step-by-step:
[tex]\[
\begin{align*}
(1 + 15) & = 16, \\
(2 + 14) & = 16, \\
(3 + 13) & = 16, \\
(4 + 12) & = 16, \\
(5 + 11) & = 16, \\
(6 + 10) & = 16, \\
(7 + 9) & = 16, \\
8 & = 8.
\end{align*}
\][/tex]
Thus, we have seven 16s and one 8:
[tex]\[
7 \times 16 + 8 = 112 + 8 = 120
\][/tex]
We verify that the sum also confirms:
[tex]\[
120
\][/tex]
Write your check in the box below:
[tex]\[
\boxed{120}
\][/tex]
