Question

find the first five partial sums of the series.
-33 - 33 - 33 - 33 - 33 - 33 - ...
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 series starts with \(-33\), so \( S_1=-33 \).

Step2: Find \( S_2 \)

The second partial sum \( S_2 \) is the sum of the first two terms. The first term is \(-33\) and the second term is also \(-33\). So \( S_2=-33 + (-33)=-66 \).

Step3: Find \( S_3 \)

The third partial sum \( S_3 \) is the sum of the first three terms. We already know the sum of the first two terms is \(-66\), and the third term is \(-33\). So \( S_3=-66+(-33) = -99 \).

Step4: Find \( S_4 \)

The fourth partial sum \( S_4 \) is the sum of the first four terms. The sum of the first three terms is \(-99\), and the fourth term is \(-33\). So \( S_4=-99+(-33)=-132 \).

Step5: Find \( S_5 \)

The fifth partial sum \( S_5 \) is the sum of the first five terms. The sum of the first four terms is \(-132\), and the fifth term is \(-33\). So \( S_5=-132+(-33)=-165 \).

Answer:

\( S_1 = -33 \)
\( S_2 = -66 \)
\( S_3 = -99 \)
\( S_4 = -132 \)
\( S_5 = -165 \)