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Chapter 14: Problem 16

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 \(a_{20}\) is \(\frac{441}{400}\).

Step by step solution

01

Identify the Given Sequence

The given sequence is represented by the formula: \[ a_n = \left(1 + \frac{1}{n}\right)^{2} \].
02

Substitute the Term Number

To find the indicated term \(a_{20}\), substitute \(n\) with 20 in the formula: \[ a_{20} = \left(1 + \frac{1}{20}\right)^{2} \].
03

Simplify Inside the Parentheses

Calculate \(1 + \frac{1}{20}\): \[ 1 + \frac{1}{20} = \frac{20}{20} + \frac{1}{20} = \frac{21}{20} \].
04

Square the Result

Square the result from the previous step: \[ \left( \frac{21}{20} \right)^{2} = \frac{21^2}{20^2} = \frac{441}{400} \].

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Key Concepts

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

Sequence Formula
In algebra, a sequence is a set of numbers arranged in a specific order, often defined by a formula.
The sequence formula tells us how to find any term in the sequence using its position number, known as 'n'.
For example, in this exercise, we are given the sequence formula \( a_n = \left(1 + \frac{1}{n}\right)^{2} \).
This means that for any given position 'n', we can find the corresponding term by plugging 'n' into the formula.
Term Substitution
Term substitution is the process of replacing 'n' with a specific value to find the desired term in the sequence.
For instance, to find the 20th term (\tex a_{20}\tex}), we substitute n with 20 in the sequence formula.
The substitution looks like this: \( a_{20} = \left(1 + \frac{1}{20}\right)^{2} \).
This step-by-step approach ensures we accurately apply the sequence formula to find the exact term.
Simplifying Fractions
Simplifying fractions involves combining or reducing fractions to their simplest form. Here鈥檚 how we simplified:\( \left(1 + \frac{1}{20}\right)^{2}\):
  • First, we solve inside the parentheses: 1 + \(\frac{1}{20}\) = \(\frac{20}{20} + \frac{1}{20} = \frac{21}{20}\).

  • Then we square the fraction: \( \left( \frac{21}{20} \right)^{2} \), which is \( \frac{21^2}{20^2} = \frac{441}{400} \).

The simplified form is easier to interpret and use.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations arranged together.
In our sequence formula \( a_n = \left(1 + \frac{1}{n}\right)^{2} \), we use basic algebraic operations:
  • Addition inside the parentheses: 1 + \frac{1}{n}\.
  • Exponentiation: Squaring the result to find the final term value.

Algebraic expressions help us generalize patterns and apply formulas systematically.

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Most popular questions from this chapter

Rewrite each sum using sigma notation. Answers may vary. $$ 1+4+9+16+25+36 $$

Find the nth, or general, term for each geometric sequence. $$ \frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}, \ldots $$

Find a formula for the sum of the first \(n\) consecutive odd numbers starting with 1: $$ 1+3+5+\cdots+(2 n-1) $$

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=0, a_{n+1}=a_{n}^{2}+3 $$

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$
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