In Exercises 61–68, use the graphs of and to find each indicated sum.∑i=15ai2+∑i=15bi2\sum_{i=1}^5a_{i}^2+\sum_{i=1}^5b_{i}^2

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 67

Capitolo 9, Problema 67

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

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

Guida verificata passo dopo passo

Step 1: Identify the values of the sequences $a_n$ and $b_n$ for $n = 1, 2, 3, 4, 5$ from the graphs. For $a_n$, read the y-values of the red points at each integer $n$ from 1 to 5. For $b_n$, do the same by reading the y-values of the red points at each integer $n$ from 1 to 5.

Step 2: Write down the values you found for each sequence. For example, if $a_1 = 4$, $a_2 = 1$, etc., and similarly for $b_n$.

Step 3: Square each of the values for both sequences. This means calculate $a_i^2$ for each $i = 1, 2, 3, 4, 5$ and $b_i^2$ for each $i = 1, 2, 3, 4, 5$.

Step 4: Sum the squared values for each sequence separately. Calculate $\sum_{i=1}^5 a_i^2$ and $\sum_{i=1}^5 b_i^2$ by adding the squared values from Step 3.

Step 5: Add the two sums together to find the final result: $\sum_{i=1}^5 a_i^2 + \sum_{i=1}^5 b_i^2$. This gives the total sum of the squares of the first five terms of both sequences.

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Sequences and Terms

A sequence is an ordered list of numbers, where each number is called a term. The term a_n represents the nth term of sequence a, and similarly b_n for sequence b. Understanding how to identify and interpret these terms from a graph is essential for evaluating sums involving sequences.

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Introduction to Sequences

Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms. For example, Σ from i=1 to 5 of a_i^2 means adding the squares of the first five terms of sequence a. This notation simplifies expressing and calculating sums of sequences.

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Interval Notation

Evaluating Terms from Graphs

To find the value of each term a_i or b_i from a graph, locate the point corresponding to the term's index on the horizontal axis and read its vertical coordinate. Squaring these values and summing them as indicated requires accurate interpretation of the graph's points.

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