Question

1 of 51 of 5 Items
Question
Calculate upper S subscript 5 baseline for the sequence defined by left-brace a subscript n baseline right-brace equals left-brace 1 plus Start Fraction 5 over 2 End Fraction n right-brace.
(1 point)
Responses

13.5
13.5

23
23

40
40

42.5

Answer 1

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:

1. For \( n = 1 \):
\[
a_1 = 1 + \frac{5}{2} \cdot 1 = 1 + 2.5 = 3.5
\]

2. For \( n = 2 \):
\[
a_2 = 1 + \frac{5}{2} \cdot 2 = 1 + 5 = 6
\]

3. For \( n = 3 \):
\[
a_3 = 1 + \frac{5}{2} \cdot 3 = 1 + 7.5 = 8.5
\]

4. For \( n = 4 \):
\[
a_4 = 1 + \frac{5}{2} \cdot 4 = 1 + 10 = 11
\]

5. 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:

1. \( 3.5 + 6 = 9.5 \)
2. \( 9.5 + 8.5 = 18 \)
3. \( 18 + 11 = 29 \)
4. \( 29 + 13.5 = 42.5 \)

Thus, the upper sum \( S_5 \) is:

\[
S_5 = 42.5
\]

The correct answer is **42.5**.