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Chapter 14: Problem 10

Find the indicated term of each sequence. $$ a_{n}=(3 n+2)^{2} ; a_{6} $$

Short Answer

Expert verified
The 6th term is 400.

Step by step solution

01

- Understand the Given Formula

The formula for the sequence is given by \(a_{n} = (3n + 2)^{2}\). This means to find the n-th term of the sequence, substitute n into the formula.
02

- Substitute the Given n Value

To find \(a_{6}\), substitute \(n = 6\) into the formula. This gives us \(a_{6} = (3 \cdot 6 + 2)^{2}\).
03

- Simplify Inside the Parentheses

First, calculate what is inside the parentheses: \(3 \cdot 6 + 2 = 18 + 2 = 20\). So the expression becomes \(a_{6} = 20^{2}\).
04

- Calculate the Square

Next, calculate the square of 20: \(20^{2} = 400\).
05

- Write the Final Answer

Thus, the 6th term of the sequence is \(a_{6} = 400\).

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Key Concepts

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

Arithmetical Sequences
An arithmetical sequence is a list of numbers where each term increases by a constant value called the common difference. This means you can predict the next term by adding this constant value to the current term. For example, in the sequence 2, 5, 8, 11, the common difference is 3. It鈥檚 important because understanding this concept helps identify patterns and calculate terms even without writing out the whole sequence.
n-th Term
The n-th term of a sequence refers to any term at position n in the sequence. To find an n-th term, we often use a formula specific to the sequence. For example, in the provided exercise, the formula is given by \(a_{n} = (3n + 2)^{2}\). This formula allows us to calculate any term of the sequence without listing all previous terms. Knowing how to use the formula is key to solving problems quickly and efficiently.
Substitution in Formulas
Substitution in formulas means replacing a variable with a specific value. This is a fundamental step in many mathematical calculations. For example, when asked to find the 6th term \(a_{6}\), we substitute 6 for n in the formula \(a_{n} = (3n + 2)^{2}\). This gives us \((3 \times 6 + 2)^{2}\). Substituting correctly ensures that we follow the correct path to the solution.
Simplification
Simplification is the process of making an expression easier to understand or solve. It involves performing arithmetic operations and reducing expressions to their simplest form. In the context of the problem, we start with \((3 \times 6 + 2)^{2}\). First, we simplify inside the parentheses: \(3 \times 6 = 18\) and then \(18 + 2 = 20\). After that, we calculate \(20^{2}\) leading to the simplified answer: 400. Each step in simplification makes the problem clearer and the final answer easier to derive.

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Most popular questions from this chapter

Straight-Line Depreciation. A company buys a color laser printer for \(\$ 5200\) on January 1 of a given year. The machine is expected to last for 8 years, at the end of which time its trade-in, or salvage, value will be \(\$ 1100 .\) If the company figures the decline in value to be the same each year, then the trade-in values, after \(t\) years, \(0 \leq t \leq 8,\) form an arithmetic sequence given by $$ a_{t}=C-t\left(\frac{C-S}{N}\right) $$ where \(C\) is the original cost of the item, \(N\) the years of expected life, and \(S\) the salvage value. a) Find the formula for \(a_{t}\) for the straight-line depreciation of the printer. b) Find the salvage value after 0 year, 1 year, 2 years, 3 years, 4 years, 7 years, and 8 years. c) Find a formula that expresses \(a_{t}\) recursively.

Rewrite each sum using sigma notation. Answers may vary. $$ \frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7} $$

Simplify. $$ \left(a_{1}+a_{n}\right)+\left(a_{1}+a_{n}\right)+\left(a_{1}+a_{n}\right) $$

Find the first five terms of each sequence. Then find \(S_{5}\). $$ a_{n}=\frac{1}{2^{n}} \log 1000^{n} $$

Review evaluating expressions and simplifying expressions. Evaluate. $$ a_{1}+(n-1) d, \text { for } a_{1}=3, n=10, \text { and } d=-2 $$
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