Question

find the first five partial sums of the series.
-5 - 7 - 3 + 13 + 47 + 105 + ...
write your answers as integers or fractions in simplest form.
s₁ =
s₂ =
s₃ =
s₄ =
s₅ =

Explanation:

Step1: Find \( S_1 \)

The first partial sum \( S_1 \) is just the first term of the series. The first term is \(-5\), so \( S_1 = -5 \).

Step2: Find \( S_2 \)

The second partial sum \( S_2 \) is the sum of the first two terms. The first term is \(-5\) and the second term is \(-7\). So \( S_2 = -5 + (-7) = -12 \).

Step3: Find \( S_3 \)

The third partial sum \( S_3 \) is the sum of the first three terms. We already know \( S_2 = -12 \), and the third term is \(-3\). So \( S_3 = S_2 + (-3) = -12 + (-3) = -15 \).

Step4: Find \( S_4 \)

The fourth partial sum \( S_4 \) is the sum of the first four terms. We know \( S_3 = -15 \), and the fourth term is \( 13 \). So \( S_4 = S_3 + 13 = -15 + 13 = -2 \).

Step5: Find \( S_5 \)

The fifth partial sum \( S_5 \) is the sum of the first five terms. We know \( S_4 = -2 \), and the fifth term is \( 47 \). So \( S_5 = S_4 + 47 = -2 + 47 = 45 \).

Answer:

\( S_1 = -5 \)
\( S_2 = -12 \)
\( S_3 = -15 \)
\( S_4 = -2 \)
\( S_5 = 45 \)