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Chapter 14: Problem 8

Find the indicated term of each sequence. $$ a_{n}=3 n+2 ; a_{8} $$

Short Answer

Expert verified
The 8th term is 26.

Step by step solution

01

Identify the General Formula

The general formula for the sequence is given by \(a_n = 3n + 2\). This formula will be used to find the specific term required.
02

Substitute the Term Number

We are asked to find the 8th term in the sequence \(a_8\). Substitute \(n = 8\) into the general formula.
03

Simplify the Expression

Substitute \(n = 8\) into the formula: \(a_8 = 3(8) + 2 = 24 + 2 = 26\). This gives us the value of the 8th term.

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Key Concepts

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

headline of the respective core concept
Understanding the process of finding a specific term in a sequence starts with identifying the general formula. The general formula provides a rule to calculate any term in the sequence by plugging in the term number. For example, in our sequence, the general formula is given by: \(a_n = 3n + 2\). This rule means that to find any term \(a_n\), you multiply the term number \(n\) by 3 and then add 2. Breaking it down step-by-step:
  • The term number \(n\) is a placeholder. It will be replaced by the specific term number you want to find.
  • The formula uses simple arithmetic operations (multiplication and addition) to generate the sequence terms from the term number \(n\).
  • Understanding the general formula is crucial because it is the starting point for finding any term in the sequence.
headline of the respective core concept
Term substitution is where you use the general formula to find a specific term in the sequence. In our exercise, we need to find the 8th term, \(a_8\). To do this, we substitute \(n = 8\) into the general formula:
  • Take the general formula \(a_n = 3n + 2\) and replace \(n\) with 8.
  • It transforms the formula to \(a_8 = 3(8) + 2\).
  • Substitution helps us narrow down the formula to a specific term by focusing on just one value of \(n\).
Substitution makes the general formula particular to the term you are interested in, thus setting the stage for the next step: simplification.
headline of the respective core concept
Sequence simplification takes the substituted formula and simplifies it to find the term's value. Using our example where the formula is transformed to \(a_8 = 3(8) + 2\):
  • First, perform the multiplication: \(3脳8 = 24\).
  • Next, add the 2: \(24 + 2 = 26\).
  • You now have the 8th term of the sequence: \(a_8 = 26\).
Simplification is the process of feeding the numeric operations to reach the final value of the term. It ensures that you derive the term clearly and correctly.

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