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Chapter 11: Problem 20

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

Short Answer

Expert verified
9680

Step by step solution

01

Understand the formula

Identify the given formula for the sequence, which is: \[ a_n = 4n^2(2n - 39) \] We need to find the term \( a_{22} \).
02

Substitute the value of n

Substitute \( n = 22 \) into the formula: \[ a_{22} = 4(22)^2(2(22) - 39) \]
03

Calculate the square

First, calculate \( 22^2 \): \[ 22^2 = 484 \]
04

Multiply by 4

Next, multiply that result by 4: \[ 4 \times 484 = 1936 \]
05

Calculate the inner term

Then, calculate the inner term \( 2(22) - 39 \): \[ 2(22) = 44 \] \[ 44 - 39 = 5 \]
06

Final multiplication

Multiply the results from steps 4 and 5: \[ 1936 \times 5 = 9680 \]

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Key Concepts

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

Algebraic Sequences
A sequence is a list of numbers in a specific order. In algebraic sequences, each term is a function of its position within the sequence. The given sequence formula is \[ a_n = 4n^2(2n - 39) \]. This formula provides the rule for finding any term, \(a_n\), in the sequence by substituting the position \(n\). Algebraic sequences are essential because they help find patterns and make predictions about future terms quickly.
Substitution
Substitution is replacing one element with another. In our exercise, we substitute the position number \(n = 22\) into our sequence formula. This means every occurrence of \(n\) in the formula will be replaced with 22. For example, \(a_{22} = 4(22)^2(2 \times 22 - 39)\). Substitution simplifies complex expressions by using known values, making calculations manageable. It's a fundamental algebraic technique.
Multiplication
Multiplication involves combining numbers to get their product. It's a basic arithmetic operation. In the sequence formula, we use multiplication multiple times: \[ 4 \times 484, \ 2 \times 22, \ \text{and}\ 1936 \times 5 \]. Each multiplication step simplifies the expression further. Multiplication in algebra often deals with combining coefficients and variables, which is crucial for expanding and simplifying expressions.
Polynomials
Polynomials consist of several terms combined using addition, subtraction, and multiplication. In this sequence, \[ 4n^2(2n - 39) \] is a polynomial. It's a second-degree polynomial because the highest power of \(n\) is 2. Polynomials are fundamental in algebra because they appear in many different types of problems, including sequence term calculations. They help in modeling various relationships and functions.

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