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Chapter 14: Problem 19

Evaluate the indicated term for each arithmetic sequence. $$ 2,4,6, \ldots ; \quad a_{24} $$

Short Answer

Expert verified
a_{24} = 48

Step by step solution

01

Identify the first term

The first term of the arithmetic sequence, denoted as \(a_1\), is the initial number in the sequence. Here, \(a_1 = 2\).
02

Determine the common difference

The common difference \(d\) is calculated by subtracting the first term from the second term. Here, \(d = 4 - 2 = 2\).
03

Use the nth-term formula for arithmetic sequences

The formula to find the nth term of an arithmetic sequence is \(a_n = a_1 + (n - 1) \times d\). Plug in the known values: \(a_{24} = 2 + (24 - 1) \times 2\).
04

Simplify the expression

Simplify the expression inside the parentheses first: \(a_{24} = 2 + 23 \times 2\). Then, perform the multiplication: \(a_{24} = 2 + 46\).
05

Calculate the final value

Add the terms to find the value of \(a_{24}\): \(a_{24} = 48\).

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Key Concepts

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

nth-term formula
The nth-term formula is crucial for finding any term in an arithmetic sequence without listing all the terms.
The general formula is \(a_n = a_1 + (n - 1) \times d\).
Here are the components:
  • \(a_n\): the nth term we want to find.
  • \(a_1\): the first term of the sequence.
  • \(d\): the common difference between terms.
  • \(n\): the position of the term in the sequence.
To use this formula, insert the values you already know. For example, to find the 24th term of the sequence \(2, 4, 6, ...\), identify the first term \(a_1 = 2\) and the common difference \(d = 2\). Then, substitute these into the formula along with \(n = 24\):
\(a_{24} = 2 + (24 - 1) \times 2\).
Simplify to get \(a_{24} = 2 + 46 = 48\).
This makes solving problems quick and efficient!
common difference
The common difference \(d\) is a key concept in arithmetic sequences.
It is the amount by which each term increases (or decreases) from the previous term.
To find it, subtract the first term from the second term:
\(d = a_2 - a_1\).
For our sequence \(2, 4, 6, ...\), the common difference is \(4 - 2 = 2\).
The common difference remains constant throughout the entire sequence.
If you know the first term and the common difference, you can generate the entire sequence.
first term
The first term of an arithmetic sequence, denoted as \(a_1\), is simply the starting point of the sequence.
It is the initial value from which all subsequent terms are derived by adding the common difference.
In the sequence \(2, 4, 6, ...\), the first term is \(a_1 = 2\).
Knowing the first term is essential because it anchors the sequence.
Combined with the common difference, it allows you to find any term using the nth-term formula.
Always start by identifying \(a_1\), as it simplifies the process of solving the sequence.

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Most popular questions from this chapter

Find the indicated term for each sequence. $$ a_{n}=(5 n-2)(3 n+1) ; \quad a_{10} $$

Find the indicated term for each sequence. $$ a_{n}=\frac{5 n-9}{3 n+8} ; \quad a_{16} $$

Solve each applied problem by writing the first few terms of a sequence. A certain car loses \(\frac{1}{2}\) of its value each year. If this car cost \(\$ 40,000\) new, what is its value at the end of 6 yr?

Solve each problem involving an ordinary annuity. At the end of each quarter, a 50 -year-old woman puts \(\$ 1200\) in a retirement account that pays \(5 \%\) interest compounded quarterly. When she reaches age \(60,\) she withdraws the entire amount and places it in a mutual fund that pays \(6 \%\) interest compounded monthly. From then on, she deposits \(\$ 300\) in the mutual fund at the end of each month. How much is in the account when she reaches age \(65 ?\)

If the given sequence is arithmetic, find the common difference d. If the sequence is not arithmetic, say so. $$ 2,-4,6,-8,10,-12, \dots $$
See all solutions

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