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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 51
Chapter 9, Problem 51

Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58.

Find a14+b12.

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1
Identify the given points for the arithmetic sequence {a} from the graph: (1, 4), (2, 7), and (3, 10). These points represent the terms a_1 = 4, a_2 = 7, and a_3 = 10.
Calculate the common difference \( d_a \) of the arithmetic sequence {a} by subtracting consecutive terms: \( d_a = a_2 - a_1 = 7 - 4 \).
Write the general formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n - 1)d_a \). Substitute \( a_1 = 4 \) and the value of \( d_a \) found in the previous step.
Since the problem asks for \( a_{14} + b_{12} \), you need to find the sequence {b} as well. However, the graph only shows sequence {a}. If sequence {b} is not given, you must find or be provided with its first term and common difference to write its nth term formula: \( b_n = b_1 + (n - 1)d_b \).
Once you have the formulas for both sequences, substitute \( n = 14 \) into the formula for {a} to find \( a_{14} \), and substitute \( n = 12 \) into the formula for {b} to find \( b_{12} \). Finally, add these two values to get \( a_{14} + b_{12} \).

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Key Concepts

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

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. This difference is called the common difference. For example, the sequence 4, 7, 10 has a common difference of 3.
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General Term Formula of an Arithmetic Sequence

The nth term of an arithmetic sequence can be found using the formula a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula allows calculation of any term without listing all previous terms.
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Using Graphs to Identify Sequence Terms

Graphs of arithmetic sequences plot term numbers (n) on the x-axis and term values (a_n) on the y-axis. By analyzing points on the graph, you can determine the first term and common difference, which helps in finding specific terms like a_14 or b_12.
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