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Chapter 13: Problem 2

For the arithmetic sequence having \(a_{n}=2 n+4,\) the term \(a_{3}=\) ____________.

Short Answer

Expert verified
10

Step by step solution

01

- Identify the Formula

The general formula for the arithmetic sequence is given as \(a_n = 2n + 4\). This formula expresses the \(n\)-th term of the sequence.
02

- Substitute the Term Number

To find the third term, substitute \(n = 3\) into the formula \(a_n = 2n + 4\).
03

- Perform the Substitution

Substitute \(n = 3\) into the formula: \(a_3 = 2(3) + 4\).
04

- Simplify the Expression

Simplify the expression: \(a_3 = 6 + 4 = 10\).
05

Conclusion

Thus, the third term \(a_3\) of the arithmetic sequence is 10.

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Key Concepts

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

nth term
In an arithmetic sequence, every term is generated using a specific formula. The formula for any term in the sequence is often called the nth term, denoted as \(a_n\).
For the given arithmetic sequence, the formula is \(a_n = 2n + 4\).
This means that to find any term \(a_n\) in the sequence, you simply plug the term number (\(n\)) into the formula. For example, if you want to find the 5th term (\(a_5\)), you substitute 5 for \(n\): \a_5 = 2(5) + 4\.
  • Step 1: Identify the term number (\(n\)). In this case, \(n = 3\) as we need to find the third term.
  • Step 2: Use the general formula provided.
  • Step 3: Substitute \(n\) into the formula and simplify to find the desired term.
substitution
Substitution is a crucial step in solving for specific terms in arithmetic sequences. Here, it involves replacing a variable (\(n\)) in the formula with a given number to compute the result.
For the given arithmetic sequence, the formula \a_n = 2n + 4\ is used to determine any term by substituting the required value of \(n\).
  • Identify the term to be found (third term, so \(n=3\)).
  • Substitute this value into the formula: \(a_3 = 2(3) + 4\).
  • Perform basic arithmetic operations to compute the value.

By this method, we determine that to find \(a_3\):
  • Replace \(n\) with 3 in \(2n + 4\).
  • It becomes \(2(3) + 4\).
  • Simplify to find the value of \(a_3\).
simplification
Simplification is the process of reducing expressions to their simplest form. It involves performing arithmetic operations, combining like terms, and applying mathematical rules efficiently.
In our exercise, after substituting \(n = 3\) into the formula, the expression \(2(3) + 4\) needs to be simplified.
  • First, multiply 2 by 3, which gives 6.
  • Then, add 6 to 4, yielding 10.

Thus, through simplification:
  • The expression \(2(3) + 4\) simplifies to \(10\).
  • The term \(a_3 = 10\).

Simplifying expressions is essential, as it makes complex problems easier to understand and solve. It involves basic arithmetic operations that should be performed sequentially to obtain the desired result.

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Most popular questions from this chapter

Give answers to the nearest thousandth. $$ \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243} $$

Prove each statement for every positive integer value of \(n .\) $${ }_{n} C_{0}=1$$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$\begin{aligned}&\left(a^{m}\right)^{n}=a^{m n}\\\&\text { (Assume that } a \text { and } m \text { are constant.) }\end{aligned}$$

The repeating decimal \(0.99999 \ldots\) can be written as the sum of the terms of a geometric sequence with \(a_{1}=0.9\) and \(r=0.1\) \(0.99999 \ldots=0.9+0.9(0.1)+0.9(0.1)^{2}+0.9(0.1)^{3}+0.9(0.1)^{4}+0.9(0.1)^{5}+\cdots\) Because \(|0.1|<1,\) this sum can be found from the formula \(S=\frac{a_{1}}{1-r} .\) Use this formula to find a more common way of writing the decimal \(0.99999 \ldots .\)

Use the binomial theorem to expand each binomial. $$ (x+r)^{5} $$
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