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Chapter 14: Problem 20

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

Short Answer

Expert verified
\( a_{12} = -29 \)

Step by step solution

01

Understand the Sequence Formula

The given formula for the sequence is \[a_{n}=-3n+7\]. It defines each term of the sequence based on the index number \( n \).
02

Substitute the Given Term Number

We need to find \( a_{12} \). Substitute \( n = 12 \) into the sequence formula: \[a_{12}=-3(12)+7\].
03

Perform the Calculation

First, calculate the product: \[-3 \times 12 = -36\]. Then, add 7 to the result: \[-36 + 7 = -29\].
04

State the Final Answer

The value of \( a_{12} \) is \( -29 \).

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Key Concepts

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

Sequence Formula
In mathematics, a sequence formula allows us to determine any term in the sequence if we know the formula and the position of the term. The formula for a sequence usually looks something like \(a_{n} = -3n + 7\). Here, \(n\) is the position or index of the term in the sequence. The formula tells us that for any index number \(n\), the corresponding term \(a_{n}\) can be calculated using the equation given. This formula makes it simple to find any term in the arithmetic sequence without listing all terms.
Index Number Substitution
To find a specific term in a sequence, you substitute the index number of that term into the sequence formula. For example, if we want to find \(a_{12}\), we substitute \(n = 12\) into the formula \[a_{n}=-3n+7\]. This involves plugging in the number 12 wherever we see \(n\) and then following the order of operations to simplify. Substituting correctly is crucial for accurate results.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers where each term after the first is generated by adding a constant to the previous term. The given example \[a_{n}=-3n+7\] is a type of arithmetic sequence. The constant difference between consecutive terms in this particular sequence is \(-3\). This difference is seen clearly in the formula as it affects \(-3n\). Arithmetic sequences are highly systematic, making calculations straightforward.

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Most popular questions from this chapter

Write each series as a sum of terms and then find the sum. $$ \sum_{i=1}^{5}(i+3) $$

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