Express each sum using summation notation. Use 11 as the lower limit of summation and ii for the index of summation. 4+42/2+43/3+⋯$$+4n/n4+4^2$/2+4^3/3$$+⋯+ 4^n/n

Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 51

9장, 문제 51

Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $4 + 4^{2} / 2 + 4^{3} / 3 + \dots + 4^{n} / n$

검증된 단계별 안내

Identify the pattern in the given sum: the first term is $4$, the second term is $\frac{4^2}{2}$, the third term is $\frac{4^3}{3}$, and so on, up to $\frac{4^n}{n}$.

Notice that each term can be written as $\frac{4^i}{i}$, where $i$ is the index of summation starting from 1 and going up to $n$.

Since the problem asks to express the sum using summation notation with lower limit 1 and index $i$, write the sum as $\sum_{i=1}^n \frac{4^i}{i} $.

This notation compactly represents the entire sum from the first term to the $n$-th term, capturing the pattern of powers of 4 divided by their respective indices.

Thus, the summation notation for the given sum is $\sum_{i=1}^n \frac{4^i}{i} $.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. This notation simplifies writing long sums and helps in analyzing series.

Index of Summation and Limits

The index of summation (commonly i) represents the variable that changes in each term of the sum. The lower limit indicates where the summation starts (here, 1), and the upper limit (n) shows where it ends. Properly setting these limits is essential for accurately expressing the sum.

General Term of a Series

The general term defines the formula for each term in the sum based on the index. In this problem, each term is of the form 4^i divided by i. Identifying this pattern allows you to write the entire sum compactly using summation notation.

Express each sum using summation notation. Use 11 as the lower limit of summation and ii for the index of summation. 4+42/2+43/3+⋯+4n/n4+4^2/2+4^3/3+⋯+ 4^n/n