Question

find the first five partial sums of the series.
16 - 64 + 256 - 1024 + 4096 - 16384 + …
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 \( 16 \), so \( S_1 = 16 \).

Step2: Find \( S_2 \)

The second partial sum \( S_2 \) is the sum of the first two terms. The first two terms are \( 16 \) and \( -64 \). So we calculate \( 16 + (-64) = 16 - 64 = -48 \). Thus, \( S_2 = -48 \).

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 \( -48 \), and the third term is \( 256 \). So we calculate \( -48 + 256 = 208 \). Thus, \( S_3 = 208 \).

Step4: Find \( S_4 \)

The fourth partial sum \( S_4 \) is the sum of the first four terms. We know the sum of the first three terms is \( 208 \), and the fourth term is \( -1024 \). So we calculate \( 208 + (-1024) = 208 - 1024 = -816 \). Thus, \( S_4 = -816 \).

Step5: Find \( S_5 \)

The fifth partial sum \( S_5 \) is the sum of the first five terms. We know the sum of the first four terms is \( -816 \), and the fifth term is \( 4096 \). So we calculate \( -816 + 4096 = 3280 \). Thus, \( S_5 = 3280 \).

Answer:

\( S_1 = 16 \)
\( S_2 = -48 \)
\( S_3 = 208 \)
\( S_4 = -816 \)
\( S_5 = 3280 \)