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Chapter 14: Problem 23

Find the indicated term for each sequence. $$ a_{n}=(n+1)(2 n+3) ; \quad a_{8} $$

Short Answer

Expert verified
The indicated term, \( a_{8} \), is 171.

Step by step solution

01

Identify the General Term Formula

The general term for the sequence is given by the formula: \[ a_{n} = (n+1)(2n+3) \]
02

Substitute the Given Value

We need to find the term when \( n = 8 \). Substitute 8 for \( n \) in the general term formula: \[ a_{8} = (8+1)(2 \times 8 + 3) \]
03

Simplify the Expression

Simplify the expression step-by-step: 1. Calculate \( 8+1 \): \[ 8+1 = 9 \]2. Calculate \( 2 \times 8 + 3 \): \[ 2 \times 8 = 16 \] \[ 16 + 3 = 19 \]3. Multiply the two results: \[ 9 \times 19 = 171 \]
04

Result

Therefore, the indicated term is: \[ a_{8} = 171 \]

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Key Concepts

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

general term formula
A general term formula is a mathematical expression that helps us find any term in a sequence. It provides a way to calculate the value of the nth term without listing all previous terms. In this exercise, the general term formula is given by:
The sequence is defined as: a_{n}=(n+1)(2n+3).
This notation means that,
for any integer value of 'n', we can plug it into the formula to find the corresponding term in the sequence. In this case, 'n' represents the position of the term in the sequence.
For instance, if we want to find the 8th term (=8), we'll substitute 'n' with 8 in the formula and simplify it to get the required term.
Let's move on to the next important concept.
substitution in sequences
Substitution in sequences involves replacing the variable in the general term formula with a specific value to find a particular term. In the given problem, we needed to find , the 8th term of the sequence. So, we substituted n=8 into the formula.
Here is how we did it:
  • Start with the general term formula:
    • a_{n}=(n+1)(2n+3).
  • Substitute the value 8 for 'n':
    • 补冲调8皑=(8+1)(2脳8+3).
  • This substitution transforms our formula into a specific expression that, when simplified, will give us the 8th term of the sequence.
By performing this substitution, we turn a general statement into a specific one that we can work with easily.
Now, let's dive into the final concept.
simplifying expressions
Simplifying expressions is a crucial step in solving mathematical problems involving sequences. Once we substitute the specific value into the general term formula, we need to simplify the resulting expression to find the exact term.
Let's continue with our example:
  • First, we substituted,补冲调8皑=(8+1)(2脳8+3).
  • We then calculate each part step-by-step:
  • Calculate inside the parentheses:8+1=9
  • Calculate the product inside the second parentheses:
    • a_{8} = 2脳8+316+3This results in 19.
  • Finally, multiply the two results:
    • 9脳19=171.
Thus, the simplified form of the expression gives us the value of the 8th term: a_{8}=171.By following these steps, anyone can simplify expressions and accurately find terms in a sequence. Simplification is just breaking down a problem into smaller, manageable steps and solving each part one at a time. This makes complex problems more approachable.

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