Question

find the first three partial sums of the series.
8 - 32 + 128 - 512 + 2048 - 8192 + …
write your answers as integers or fractions in simplest form.
s₁ =
s₂ =
s₃ =

Explanation:

Step1: Find \( S_1 \)

The first partial sum \( S_1 \) is just the first term of the series. The first term is 8, so \( S_1 = 8 \).

Step2: Find \( S_2 \)

The second partial sum \( S_2 \) is the sum of the first two terms. The first term is 8 and the second term is -32. So we calculate \( 8 + (-32) = 8 - 32 = -24 \), thus \( S_2 = -24 \).

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 -24, and the third term is 128. So we calculate \( -24 + 128 = 104 \), hence \( S_3 = 104 \).

Answer:

\( S_1 = 8 \)
\( S_2 = -24 \)
\( S_3 = 104 \)