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Chapter 14: Problem 14

Find the indicated term of each sequence. $$ a_{n}=4 n^{2}(2 n-39) ; a_{22} $$

Short Answer

Expert verified
The 22nd term is 9680.

Step by step solution

01

Identify the given sequence formula

The given sequence formula is \( a_{n}=4n^{2}(2n-39) \).This formula will be used to find the 22nd term of the sequence.
02

Substitute 22 into the formula

To find the 22nd term, substitute \( n = 22 \) into the formula. So, we need to calculate \( a_{22}=4(22)^{2}(2(22)-39) \).
03

Calculate the inner terms

First, calculate \( 22^2 \): \( 22^2 = 484 \).Next, calculate \( 2(22) - 39 \): \( 2(22) - 39 = 44 - 39 = 5 \).
04

Multiply the results

Now, multiply the results: \( 4 \times 484 \times 5 \).
05

Final calculation

Calculate the result: \( 4 \times 484 = 1936 \).Finally, \( 1936 \times 5 = 9680 \).So, \( a_{22} = 9680 \).

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Key Concepts

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

sequence formula
In mathematics, a sequence formula defines the pattern or rule that generates the terms of a sequence. The given formula for this exercise is: \[ a_{n} = 4n^2(2n - 39) \]This formula tells us how to find any term in the sequence by plugging in the value of \( n \). For example, for the 22nd term, we substitute \( n = 22 \) in the formula.
substitution
Substitution simply means replacing a variable with a given number or another expression. In this exercise, to find the 22nd term, we substitute 22 for \( n \) in the sequence formula: \[ a_{22} = 4(22)^2(2(22) - 39) \]This step allows us to transform the formula with a specific value and helps in calculating the desired term.
exponentiation
Exponentiation is the mathematical operation involving two numbers, the base and the exponent. For example, in \( 22^2 \), 22 is the base and 2 is the exponent. It means 22 is multiplied by itself: \[ 22^2 = 22 \times 22 = 484 \]In the step-by-step solution, we calculate \( 22^2 \) before using it in further operations.
multiplication
Multiplication involves taking two numbers and calculating their product. In our formula, we perform several multiplication operations. After computing \( 22^2 \ = 484 \) and \( 2(22) - 39 = 5 \), we multiply these results with 4 according to the given sequence: \[ 4 \times 484 \times 5 \]The rule is to perform multiplication step-by-step to keep calculations manageable.
problem-solving steps
To tackle any algebra problem effectively, follow these systematic steps:
  • Identify the given formula or rule.
  • Substitute the given values into the formula.
  • Perform operations in the correct order (exponentiation, then multiplication).
  • Check each step to ensure accuracy.
For this specific problem, these steps help determine the 22nd term of the sequence: 1. Given formula: \[ a_{n} = 4n^2(2n - 39) \]2. Substitute: \[ n = 22 \]3. Calculate exponent and inner terms: \[ 22^2 = 484 \] and \[ 2(22) - 39 = 5 \]4. Multiply: \[ 4 \times 484 = 1936 \] and then \[ 1936 \times 5 = 9680 \].

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