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Chapter 12: Problem 13

Find the indicated term for each sequence. $$ a_{n}=-9 n+2 ; \quad a_{8} $$

Short Answer

Expert verified
The 8th term is -70.

Step by step solution

01

Understand the Problem

We are given an arithmetic sequence defined by the formula for its nth term: \(a_{n} = -9n + 2\) and we need to find the 8th term (\(a_{8}\)).
02

Substitute n with 8

To find \(a_{8}\), substitute \(n = 8\) into the formula: \(a_{8} = -9(8) + 2\).
03

Perform the Calculation

First, calculate \(-9 \times 8\): \(-9 \times 8 = -72\). Then add 2 to the result: \(-72 + 2 = -70\).
04

State the Answer

Thus, the 8th term of the sequence is \(-70\).

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Key Concepts

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

nth term formula
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The nth term formula of an arithmetic sequence is one of the basic tools to work with such sequences. It allows us to find any term in the sequence without listing all the terms.
The formula for the nth term of an arithmetic sequence is usually given as:

a_{n} = a_1 + (n-1)d
where:
  • \(a_{n}\) is the nth term,
  • \(a_1\) is the first term,
  • \(n\) is the position of the term in the sequence,
  • \(d\) is the common difference between the terms.
In our given problem, the formula is a bit different:
\(a_{n} = -9n + 2\).
This is a valid nth term formula, representing an arithmetic sequence with a different format. Here, the common difference and the first term are embedded in the formula given.
substitution in sequences
To find a specific term in an arithmetic sequence using the nth term formula, we need to use the process of substitution. This means replacing the variable \(n\) in the formula with the term number we are interested in.
For instance, in our exercise, we are asked to find the 8th term in the sequence defined by: \(a_{n} = -9n + 2\).
Let's substitute 8 for \(n\):
  • \(a_{8} = -9(8) + 2\)
Substitution is a straightforward step. It's simply about replacing \(n\) with the desired term position number so that we can then carry out the arithmetic operations.
sequence calculation
Once we have substituted the value of \(n\) into our nth term formula, the next step is to perform the sequence calculation. This involves doing basic arithmetic operations to solve for the term. Let's break it down:
  • We substituted \(n = 8\) into the formula: \(a_{8} = -9(8) + 2\).
  • Next, we need to do the multiplication: \(-9 \times 8\).
  • This gives us: \(-72\).
  • Then we add the constant term (which is 2): \(-72 + 2\).
  • Finally, we get: \(-70\).
So, the 8th term of our sequence is \(-70\). Every step in the calculation needs to be done carefully to avoid any mistakes. Double-check each arithmetic operation to ensure accuracy.

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Most popular questions from this chapter

Find a general term for each geometric sequence. $$ -2,-6,-18, \dots $$

Evaluate each expression. $$ 8 ! $$

Solve each problem involving an ordinary annuity. B. G. Thompson puts \(\$ 1000\) in a retirement account at the end of each quarter \(\left(\frac{1}{4} \text { of a year }\right)\) for 15 yr. If the account pays \(4 \%\) annual interest compounded quarterly, how much will be in the account at that time?

Solve each applied problem by writing the first few terms of a sequence. Horacio Loschak borrows \(\$ 1000\) and agrees to pay \(\$ 100\) plus interest of \(1 \%\) on the unpaid balance each month. Find the payments for the first six months and the remaining debt at the end of that period.

Write the first five terms of each geometric sequence. $$ a_{1}=2, r=3 $$
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