In Exercises 61–68, use the graphs of and to find each indicated sum.∑i=15(ai2+1)\sum_{i=1}^5(a_{i}^2+1)

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 61

9장, 문제 61

In Exercises 61–68, use the graphs of and to find each indicated sum.

$\sum_{i = 1}^{5} (a_{i}^{2} + 1)$

검증된 단계별 안내

Identify the values of the sequence $ a_n $ from the graph for $ n = 1 $ to $ n = 5 $. From the graph, the points are: $ a_1 = -2 $, $ a_2 = 0 $, $ a_3 = 2 $, $ a_4 = 4 $, and $ a_5 = 6 $.

Write the expression for the sum you need to find: $ \sum_{i=1}^5 (a_i^2 + 1) $. This means for each $ i $ from 1 to 5, you square the value of $ a_i $, then add 1, and finally sum all these results.

Calculate each term inside the sum separately: For each $ i $, compute $ a_i^2 + 1 $. For example, for $ i=1 $, calculate $ (-2)^2 + 1 $, for $ i=2 $, calculate $ 0^2 + 1 $, and so on.

After finding each term $ a_i^2 + 1 $, add all these values together to get the total sum.

Write the final sum as $ (a_1^2 + 1) + (a_2^2 + 1) + (a_3^2 + 1) + (a_4^2 + 1) + (a_5^2 + 1) $ and simplify if needed.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Sequences and Terms

A sequence is an ordered list of numbers where each number is called a term, denoted as a_n. Understanding how to identify and interpret terms from a graph is essential, as each point corresponds to a term's value at a specific position n.

Summation Notation (Sigma Notation)

Summation notation, represented by the Greek letter Σ, is a concise way to express the sum of a sequence of terms. It includes an index of summation, lower and upper limits, and the general term to be summed, allowing efficient calculation of sums like Σ from i=1 to 5.

Evaluating Expressions Involving Sequence Terms

To find sums involving expressions like (a_i^2 + 1), you must first determine each term a_i from the graph, then square it, add 1, and finally sum all these values over the given range. This process combines understanding of sequences, algebraic operations, and summation.

In Exercises 61–68, use the graphs of and to find each indicated sum.∑i=15(ai2+1)\(\sum\)_{i=1}^5(a_{i}^2+1)