To find the upper sum \( S_5 \) for the sequence defined by \( { a_n } = { 1 + \frac{5}{2} n } \), we first need to determine the values of the sequence from \( n = 1 \) to \( n = 5 \).
Calculating the first five terms:
-
For \( n = 1 \): \[ a_1 = 1 + \frac{5}{2} \cdot 1 = 1 + 2.5 = 3.5 \]
-
For \( n = 2 \): \[ a_2 = 1 + \frac{5}{2} \cdot 2 = 1 + 5 = 6 \]
-
For \( n = 3 \): \[ a_3 = 1 + \frac{5}{2} \cdot 3 = 1 + 7.5 = 8.5 \]
-
For \( n = 4 \): \[ a_4 = 1 + \frac{5}{2} \cdot 4 = 1 + 10 = 11 \]
-
For \( n = 5 \): \[ a_5 = 1 + \frac{5}{2} \cdot 5 = 1 + 12.5 = 13.5 \]
Now, we have the first five terms of the sequence:
- \( a_1 = 3.5 \)
- \( a_2 = 6 \)
- \( a_3 = 8.5 \)
- \( a_4 = 11 \)
- \( a_5 = 13.5 \)
To calculate the upper sum \( S_5 \) using these terms, we simply add them together:
\[ S_5 = a_1 + a_2 + a_3 + a_4 + a_5 \] \[ S_5 = 3.5 + 6 + 8.5 + 11 + 13.5 \] Calculating this step-by-step:
- \( 3.5 + 6 = 9.5 \)
- \( 9.5 + 8.5 = 18 \)
- \( 18 + 11 = 29 \)
- \( 29 + 13.5 = 42.5 \)
Thus, the upper sum \( S_5 \) is:
\[ S_5 = 42.5 \]
The correct answer is 42.5.