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Chapter 11: Problem 14

Find the indicated term of each sequence. $$ a_{n}=3 n-4 ; a_{8} $$

Short Answer

Expert verified
The 8th term is 20.

Step by step solution

01

- Understand the given sequence formula

The formula given for the sequence is \[ a_n = 3n - 4 \]where each term in the sequence can be found by plugging in the value of \( n \) into the formula.
02

- Identify the term to find

The problem asks for the 8th term of the sequence, which means we need to find the value when \( n = 8 \).
03

- Plug into the formula

Substitute \( n = 8 \) into the formula: \[ a_8 = 3 \times 8 - 4 \]
04

- Calculate the result

Next, perform the multiplication and subtraction: \[ a_8 = 24 - 4 = 20 \]
05

- Finalize your answer

Therefore, the 8th term of the sequence is 20.

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Key Concepts

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

term calculation
Calculating a term in a sequence means finding a specific value in a list of numbers. This involves using the formula provided to calculate each term. Each term is identified by its position, known as 'n'.

For example, in the sequence formula given: \(a_n = 3n - 4\), 'n' represents the position. The term calculation process involves:
  • Identifying the position 'n' you need.
  • Substituting 'n' into the formula.
  • Performing the arithmetic operations to find the term value.
In our exercise, to find the 8th term:
1. Identify the term: 8th term, so n = 8.
2. Substitute into the formula: \(a_8 = 3 * 8 - 4\).
3. Calculate: \(24 - 4 = 20\). The 8th term is 20.
sequence formula
A sequence formula is a mathematical expression defining the pattern of the terms in a sequence. It allows you to find any given term without having to list all preceding terms.

Consider the sequence formula: \(a_n = 3n - 4\). It tells us how each term is related to its position 'n'. This specific formula shows that each term is calculated by multiplying 'n' by 3 and then subtracting 4.

Using this formula, you can generate as many sequence terms as you need. For example:
  • 1st term: \(a_1 = 3 * 1 - 4 = -1\)
  • 2nd term: \(a_2 = 3 * 2 - 4 = 2\)
  • 3rd term: \(a_3 = 3 * 3 - 4 = 5\)
The beauty of knowing the sequence formula is that you can directly find any term by just plugging in the value of 'n'.
plugging in values
The process of plugging in values involves substituting a specific number into the sequence formula. This allows us to find the exact value for that term.

For our exercise, we need the 8th term.

Here's how to plug in the values:
  • Start with the sequence formula: \(a_n = 3n - 4\)
  • Identify 'n': For the 8th term, n = 8.
  • Substitute 8 for 'n': \(a_8 = 3 * 8 - 4\)
  • Perform the arithmetic: \(24 - 4 = 20\)
So, after plugging in the values, the 8th term is calculated to be 20. This method removes the guesswork and ensures accuracy for any term in the sequence.

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Most popular questions from this chapter

Write out and evaluate each sum. $$ \sum_{k=0}^{5}\left(k^{2}-3 k+4\right) $$

Find the indicated term for each geometric sequence. $$\frac{7}{623}, \frac{-7}{125}, \frac{7}{25}, \ldots ; \text { the } 13 \text { th term }$$

use the formula for \(S_{n}\) to find the indicated sum for each geometric series. $$S_{6} \text { for } 16-8+4-\cdots$$

Multiply: \(\left(x^{2}+2 x y+y^{2}\right)\left(x^{2}+2 x y+y^{2}\right)^{2}(x+y)\)

Find the indicated term for each geometric sequence. \(2,2 \sqrt{2}, 4, \dots ;\) the 8 th term
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