Sure, let's solve the given question step-by-step.
### (a) Find the value of [tex]\( T \)[/tex] when [tex]\( n = 15 \)[/tex]
We are given the formula:
[tex]\[
T = \frac{n(n+1)}{2}
\][/tex]
Substitute [tex]\( n = 15 \)[/tex] into the formula:
[tex]\[
T = \frac{15(15+1)}{2}
\][/tex]
Simplify inside the parentheses:
[tex]\[
15 + 1 = 16
\][/tex]
Now substitute back into the equation:
[tex]\[
T = \frac{15 \times 16}{2}
\][/tex]
Calculate the product:
[tex]\[
15 \times 16 = 240
\][/tex]
Now divide by 2:
[tex]\[
T = \frac{240}{2} = 120
\][/tex]
So, the value of [tex]\( T \)[/tex] when [tex]\( n = 15 \)[/tex] is:
[tex]\[
T = 120
\][/tex]
### (b) Show a check of your answer.
To verify the answer, we can use a different method provided:
We are to check whether:
[tex]\[
7 \times 16 + 8 = 120
\][/tex]
First, calculate [tex]\( 7 \times 16 \)[/tex]:
[tex]\[
7 \times 16 = 112
\][/tex]
Then add 8:
[tex]\[
112 + 8 = 120
\][/tex]
Thus, our check confirms the original calculation:
[tex]\[
120 = 120
\][/tex]
So, both the initial calculation and the verification step agree on [tex]\( T \)[/tex] being [tex]\( 120 \)[/tex].
