In Exercises 61–68, use the graphs of and to find each indicated sum.∑i=15(ai+3bi)\sum_{i=1}^5(a_{i}+3b_{i})

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 64

Capitolo 9, Problema 64

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

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

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 points at $n=1, 2, 3, 4, 5$. For $b_n$, do the same for the corresponding $n$ values.

Step 2: Write down the values explicitly. For example, $a_1 = \text{value from graph}$, $a_2 = \text{value from graph}$, and so on, similarly for $b_1, b_2, b_3, b_4, b_5$.

Step 3: Calculate each term inside the summation $a_i + 3b_i$ for $i = 1, 2, 3, 4, 5$. This means multiply each $b_i$ by 3 and then add the corresponding $a_i$.

Step 4: Sum all the calculated terms from Step 3 to find $\sum_{i=1}^5 (a_i + 3b_i)$. This involves adding the five values obtained for each $i$.

Step 5: Write the final expression for the sum, showing the addition of all terms explicitly, but do not compute the numerical total as per instructions.

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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 terms from graphs is essential for working with sequences.

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Introduction to Sequences

Summation Notation (Sigma Notation)

Summation notation, represented by the Greek letter sigma (Σ), is a concise way to express the sum of a sequence of terms. For example, Σ from i=1 to 5 of (a_i + 3b_i) means adding the values of a_i plus three times b_i for i = 1 through 5.

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Interval Notation

Using Graphs to Extract Sequence Values

Graphs of sequences plot term indices on the x-axis and term values on the y-axis. To find specific terms like a_i or b_i, locate the point at x = i and read the corresponding y-value. This skill is crucial for evaluating sums involving terms from graphical data.

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