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Chapter 13: Problem 24

Find the indicated term for each sequence. $$ a_{n}=(5 n-2)(3 n+1) ; \quad a_{10} $$

Short Answer

Expert verified
1488

Step by step solution

01

Identify the Given Sequence Formula

The sequence formula provided is \(a_{n}=(5n-2)(3n+1)\). This formula will be used to find the indicated term.
02

Substitute the Given Term

Substitute \(n=10\) into the formula to find \(a_{10}\). Therefore, calculate \((5 \cdot 10 - 2)(3 \cdot 10 + 1)\).
03

Simplify the Expression Inside the Parentheses

Calculate \(5 \cdot 10 - 2 = 50 - 2 = 48\) and \(3 \cdot 10 + 1 = 30 + 1 = 31\).
04

Multiply the Results

Now, multiply the two simplified terms: \(48 \cdot 31\).
05

Perform the Multiplication

Calculate \(48 \cdot 31 = 48 \cdot (30 + 1) = 48 \cdot 30 + 48 \cdot 1 = 1440 + 48 = 1488\).

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Key Concepts

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

sequence formula
A sequence formula is a rule that describes the terms of a sequence. In this exercise, the sequence is defined by the formula \(a_{n}=(5n-2)(3n+1)\). This formula combines both linear expressions in terms of \(n\). To find any term in the sequence, you simply replace \(n\) with the term number you need.
For example, if you want to find \(a_{2}\), substitute \(n=2\) into the sequence formula.
substitute term
Substituting the term means to replace the variable \(n\) with the specific value you are interested in. In this exercise, we need to find the 10th term of the sequence, which is \(a_{10}\).
So, we substitute \(n = 10\) into the sequence formula \(a_{n}=(5n-2)(3n+1)\). This gives us \(a_{10} = (5 \times 10 - 2)(3 \times 10 + 1)\).
simplify expression
Simplifying expressions involves performing the arithmetic operations inside parentheses first. From the substitution step, we obtained \( (5 \times 10 - 2)(3 \times 10 + 1) \). Now, we simplify within the parentheses:
  • Calculate \(5 \times 10\) which is 50, then subtract 2 to get 48.
  • Calculate \(3 \times 10\) which is 30, then add 1 to get 31.
So, the expression simplifies to \(48 \times 31\).
multiplication
Now we need to multiply the simplified terms to find the final result of \(a_{10}\). We have \(48 \times 31\). An easy way to multiply is to use the distributive property:
  • First, break 31 into 30 + 1: \(48 \times (30 + 1)\).
  • Then, multiply 48 by 30: \(48 \times 30 = 1440\).
  • Next, multiply 48 by 1: \(48 \times 1 = 48\).
  • Finally, add these products together: 1440 + 48 = 1488.
So, \(a_{10} = 1488\).

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