QuestionJuly 20, 2025

Write the sum using summation notation. There may be multiple representations. Use i as the index of summation. -(1)/(4)+(1)/(16)-(1)/(64)+(1)/(256)-(1)/(1024) We can write the sum as sum _(i=1)^5square

Write the sum using summation notation. There may be multiple representations. Use i as the index of summation. -(1)/(4)+(1)/(16)-(1)/(64)+(1)/(256)-(1)/(1024) We can write the sum as sum _(i=1)^5square
Write the sum using summation notation. There may be multiple representations. Use i as the index of summation. -(1)/(4)+(1)/(16)-(1)/(64)+(1)/(256)-(1)/(1024) We can write the sum as sum _(i=1)^5square

Solution
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Answer

\sum_{i=1}^{5} \frac{(-1)^{i+1}}{4^i} Explanation 1. Identify the Pattern The sequence is -\frac{1}{4}, \frac{1}{16}, -\frac{1}{64}, \frac{1}{256}, -\frac{1}{1024}. Each term alternates in sign and can be expressed as (-1)^{i+1} for alternating signs. 2. Determine the General Term The denominators are powers of 4: 4^1, 4^2, 4^3, 4^4, 4^5. Thus, each term can be written as \frac{(-1)^{i+1}}{4^i}.

Explanation

1. Identify the Pattern<br /> The sequence is $-\frac{1}{4}, \frac{1}{16}, -\frac{1}{64}, \frac{1}{256}, -\frac{1}{1024}$. Each term alternates in sign and can be expressed as $(-1)^{i+1}$ for alternating signs.<br />2. Determine the General Term<br /> The denominators are powers of 4: $4^1, 4^2, 4^3, 4^4, 4^5$. Thus, each term can be written as $\frac{(-1)^{i+1}}{4^i}$.
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