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

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 63

Chapter 9, Problem 63

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

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

Verified step by step guidance

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 these $n$-values. For $b_n$, do the same.

Write down the values explicitly: $a_1, a_2, a_3, a_4, a_5$ and $b_1, b_2, b_3, b_4, b_5$.

Form the expression inside the summation for each $i$ from 1 to 5: calculate $2a_i + b_i$ for each $i$.

Sum all the values obtained in the previous step: $\sum_{i=1}^5 (2a_i + b_i) = (2a_1 + b_1) + (2a_2 + b_2) + (2a_3 + b_3) + (2a_4 + b_4) + (2a_5 + b_5)$.

This sum will give the final result for the indicated sum. You can now substitute the values and perform the arithmetic to find the answer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequences and Terms

A sequence is an ordered list of numbers where each number is called a term. The term a_n represents the value of the sequence at position n. Understanding how to read and interpret terms from a graph is essential for evaluating sums involving sequences.

Summation Notation

Summation notation (∑) is a concise way to represent the sum of a sequence of terms. The index i runs from the lower limit to the upper limit, and the expression inside the summation is evaluated for each i and then added together. This notation is key to solving problems involving sums of sequences.

Combining Sequences in Sums

When summing expressions involving multiple sequences, such as 2a_i + b_i, each term from the sequences must be evaluated at the same index i, multiplied or added as indicated, and then summed over the specified range. This requires careful substitution and arithmetic using the values from the graphs.

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Textbook Question

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

$\sum_{i = 1}^{5} (a_{i} + 3 b_{i})$