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Chapter 12: Problem 14

Find the indicated term for each sequence. $$ a_{n}=3 n-7 ; \quad a_{12} $$

Short Answer

Expert verified
The 12th term is 29.

Step by step solution

01

Identify the given sequence formula

The sequence formula is given by \( a_n = 3n - 7 \). This formula will be used to find the specific term requested.
02

Substitute the term number into the formula

We need to find the 12th term of the sequence, so substitute \( n = 12 \) into the formula: \( a_{12} = 3(12) - 7 \).
03

Perform the calculation

First, perform the multiplication: \( 3 \times 12 = 36 \). Then, subtract 7 from the result: \( 36 - 7 = 29 \).
04

State the final answer

The 12th term of the sequence is \( a_{12} = 29 \).

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Key Concepts

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

algebra
Algebra is a fundamental aspect of mathematics. It involves working with symbols and variables to express mathematical relationships.
In sequences, algebraic expressions often define the relationship between the position and the value of terms.
Understanding these expressions helps us calculate specific terms easily.
sequence formula
A sequence formula is a mathematical expression that defines the pattern of a sequence.
In our example, the sequence formula is given by \(a_{n} = 3n - 7\).
This formula tells us how to calculate the nth term using the position number (n).
These formulas are essential because they provide a consistent way to find any term in the sequence.
substitute and solve
To find a specific term in a sequence, we substitute the position number into the sequence formula.
For example, we need to find the 12th term in the sequence defined by \(a_{n} = 3n - 7\).
We substitute \(n = 12\) into the formula, resulting in \(a_{12} = 3(12) - 7\).
With substitution, we use algebra to simplify and solve for the term.
calculation steps
The calculation steps involve performing arithmetic operations after substitution.
For instance, substituting \(n = 12\) in \(a_{n} = 3n - 7\) gives us \(a_{12} = 3(12) - 7\).
First, multiply 3 by 12 to get 36.
Then, subtract 7 from 36 to get 29.
So, the 12th term of the sequence is 29.
It is crucial to follow these steps accurately to ensure the correct result.

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Most popular questions from this chapter

In a certain community, the consumption of electricity has increased about \(6 \%\) per yr. (a) If a community uses 1.1 billion units of electricity now, how much will it use 5 yr from now? (b) Find the number of years it will take for the consumption to double.

Evaluate the number of terms in each arithmetic sequence. $$ 2, \frac{3}{2}, 1, \frac{1}{2}, \ldots,-5 $$

Find a general term \(a_{n}\) for the given terms of each sequence. $$ -8,-16,-24,-32, \dots $$

Find a general term for each geometric sequence. $$ -3, \frac{3}{2},-\frac{3}{4}, \ldots $$

Evaluate \(S_{6}\) for each arithmetic sequence. $$ a_{1}=7, d=-3 $$
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