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Chapter 11: Problem 12

Find the indicated term of the given sequence. $$a_{n}=(3 n-4)(2 n+5) ; a_{7}$$

Short Answer

Expert verified
323

Step by step solution

01

Identify the given formula

The given sequence is defined by the formula: \[a_{n}=(3n-4)(2n+5)\] and we need to find the 7th term, denoted as \(a_{7}\).
02

Substitute the given value of n

Substitute \(n = 7\) into the sequence formula: \[a_{7} = (3(7) - 4)(2(7) + 5)\]
03

Simplify inside the parentheses

Calculate the expressions inside the parentheses: \[3(7) - 4 = 21 - 4 = 17\] \[2(7) + 5 = 14 + 5 = 19\]
04

Multiply the results

Multiply the simplified expressions to find \(a_{7}\): \[(17)(19) = 323\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indicated Term
In sequence problems, the term we are asked to find is known as the indicated term. For example, in the sequence given by the formula \(a_{n} = (3n - 4)(2n + 5)\), the indicated term is \(a_{7}\). This means we need to find the seventh term of the sequence. Knowing how to find and work with indicated terms is essential in understanding sequences.
Substitution
Substitution is a mathematical technique where we replace a variable with a specific value. In our problem, we substitute \(n = 7\) into the formula \(a_{n} = (3n - 4)(2n + 5)\). This step is crucial because it allows us to find the specific term of the sequence. After substitution, our expression looks like this: \(a_{7} = (3(7) - 4)(2(7) + 5)\). This means we replace all 'n' with '7,' transforming the sequence into a solvable expression.
Simplification
Simplification involves breaking down complex expressions into simpler forms. After substituting \(n\) in sequence, the next step is simplification. For our example, we begin with \(a_{7} = (3(7) - 4)(2(7) + 5)\): this involves simplifying the expressions within the parentheses. We simplify:
  • \(3(7) - 4\) turns into \(21 - 4\), which equals \(17\).
  • \(2(7) + 5\) turns into \(14 + 5\), which equals \(19\).
These steps reduce the complexity of the expression.
Multiplication
Multiplication is the final step in computing the indicated term. Once we have simplified the expressions inside the parentheses, we multiply them together. In this sequence problem, the simplified terms are \(17\) and \(19\). We then calculate their product like this: \((17)(19) = 323\). The result, \(323\), is the indicated term, meaning \(a_{7} = 323\). Understanding multiplication and being able to execute it accurately is fundamental in solving sequence problems.

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Most popular questions from this chapter

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