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Chapter 11: Problem 21

Find the indicated term of each sequence. $$ a_{n}=\left(1+\frac{1}{n}\right)^{2} ; a_{20} $$

Short Answer

Expert verified
The indicated term \( a_{20} = \frac{441}{400} \)

Step by step solution

01

- Understand the given formula

The sequence is given by the formula: \[ a_{n}=\bigg(1+\frac{1}{n}\bigg)^{2} \]This defines each term in the sequence based on the value of \( n \). We need to find the term \( a_{20} \).
02

- Substitute the value of n

To find \( a_{20} \), substitute \( n = 20 \) into the formula: \( a_{20} = \bigg(1 + \frac{1}{20}\bigg)^{2} \)
03

- Simplify inside the parentheses

Simplify the fraction inside the parentheses:\( 1 + \frac{1}{20} = \frac{20}{20} + \frac{1}{20} = \frac{21}{20} \)So the expression becomes: \( a_{20} = \bigg(\frac{21}{20}\bigg)^{2} \)
04

- Square the fraction

Square the fraction \( \frac{21}{20} \):\[ \bigg( \frac{21}{20} \bigg)^{2} = \frac{21^2}{20^2} = \frac{441}{400} \]Thus, \( a_{20} = \frac{441}{400} \)

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Key Concepts

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

sequence term calculation
When working with sequences in intermediate algebra, understanding how to calculate specific terms is essential. A sequence is essentially an ordered list of numbers that follow a particular rule. The general term of the sequence is usually defined by a formula, such as \(a_n = \bigg(1 + \frac{1}{n}\bigg)^2\). To find a specific term in the sequence, you need to substitute the given term number into the formula. For example, if you want to find the 20th term \(a_{20}\), you directly substitute 20 into the formula.
substitution method
The substitution method involves replacing a variable with a specific value. In this sequence, the term we need to find is \(a_{20}\).
To calculate this, we substitute \(n = 20\) into the sequence formula:
\[a_{20} = \bigg(1 + \frac{1}{20}\bigg)^2\].
Substitution is a straightforward yet powerful tool used in various algebraic problems. It helps us evaluate expressions by replacing variables with their numerical values.
fraction simplification
Simplifying fractions is a key skill in algebra. When substituting values into the sequence formula, you often end up with fractions. For instance, substituting \(n = 20\) into our sequence, we get:
\[1 + \frac{1}{20} = \frac{20}{20} + \frac{1}{20}\]
Combining these fractions involves finding a common denominator:
\[\frac{20}{20} + \frac{1}{20} = \frac{21}{20}\]
Simplifying fractions often makes it easier to work with them in subsequent mathematical steps.
squaring fractions
Squaring fractions is a basic algebraic operation where you multiply the fraction by itself. In our example, after simplifying the fraction inside the parentheses, we get \(\frac{21}{20}\). To find the value of \(a_{20}\), we need to square this fraction:
\[a_{20} = \bigg(\frac{21}{20}\bigg)^2\].
This means:
\[(\frac{21}{20})^2 = \frac{21^2}{20^2} = \frac{441}{400}\].
Understanding how to square fractions correctly is crucial for finding terms in sequences and solving other algebraic problems.

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Most popular questions from this chapter

During the 2015 season, Miguel Cabrera of the Detroit Tigers had a batting average of \(0.338 .\) In that season, if someone were to randomly select 5 of his "at-bats," the probability of Cabrera's getting exactly 3 hits would be the 3 rd term of the binomial expansion of \((0.338+0.662)^{5} .\) Find that term and use a calculator to estimate the probability.

Perform the indicated operation and, if possible, simplify. $$ \frac{y^{3}-y}{3 y+1} \div \frac{y^{2}}{9 y+3} $$

Solve. $$|2 x+5|<6$$

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