Sequences and Series

How to find the nth term of an arithmetic sequence?

Quick answer

You calculate any term in an arithmetic sequence using the formula a_n = a_1 + (n - 1)d. Simply identify the first term (a_1) and determine the common difference (d) by subtracting any term from the term that follows it. Then, substitute the target term number for n and simplify the expression.

The topic explained

The general term formula

An arithmetic sequence is an ordered list of numbers where the difference between consecutive terms is always constant. Calculating the nth term allows you to find any value in the pattern without writing out every single step.

The standard formula is a_n = a_1 + (n - 1)d. In this rule, a_n represents the value of the nth term, a_1 is the first term, n is the position number, and d is the common difference. Because you start at the first term, you add the common difference one fewer time than the term number, which is why the formula multiplies d by (n - 1).

Step-by-step method with real numbers

Consider the sequence 7, 11, 15, 19, ... and suppose you want to find the 25th term. First, identify the starting term: a_1 = 7. Next, subtract the first term from the second term to find the common difference: d = 11 - 7 = 4.

Now plug these numbers and n = 25 into the formula: a_25 = 7 + (25 - 1)(4). This simplifies to 7 + (24)(4) = 7 + 96 = 103. The 25th term of the sequence is 103.

Finding the number of terms and avoiding errors

You can also use this formula in reverse to count how many terms exist in a finite sequence. For example, if a sequence begins 5, 8, 11, ... and ends at 62, set a_n = 62. Write 62 = 5 + (n - 1)(3). Subtract 5 to get 57 = 3(n - 1), divide by 3 to get 19 = n - 1, and add 1 to find n = 20 terms.

A common mistake is using n instead of (n - 1), which adds an extra step of difference. Another common trip is subtracting terms backwards when the sequence decreases. In the sequence 20, 15, 10, ..., the difference is 15 - 20 = -5, not positive 5.

Free printable worksheet

Finding Terms in Arithmetic Sequences Worksheet

Designed for 11th grade math students. Calculate the requested term or solve for the missing variable in each sequence.

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Finding Terms in Arithmetic Sequences WorksheetName:

Find the common difference d for the sequence: 4, 10, 16, 22, ...
Write the general term formula a_n for the sequence: 3, 8, 13, 18, ...
Find the 15th term of the sequence: 12, 9, 6, 3, ...
Find the 40th term of the sequence: -5, 1, 7, 13, ...
A finite sequence is given by 8, 15, 22, ..., 141. How many terms are in this sequence?
The 4th term of an arithmetic sequence is 19 and the 9th term is 44. Find the first term a_1 and the common difference d.

Common questions

What is the difference between an arithmetic sequence and a geometric sequence?

An arithmetic sequence adds or subtracts a fixed constant called the common difference to reach the next term. A geometric sequence multiplies each term by a fixed ratio to reach the next term.

Can the common difference d be a fraction or a negative number?

Yes. A negative common difference means the sequence values decrease over time. A fractional common difference means each step changes the value by a rational fraction.

How do you find the first term if you only know two later terms?

Set up a system of two equations using a_n = a_1 + (n - 1)d. Subtract the two equations to solve for the common difference d first, then substitute d back into either equation to calculate a_1.

How long does it take an 11th grade student to master arithmetic sequences?

Most high school students grasp identifying the pattern and applying the basic nth-term formula within one to two focused class periods, followed by targeted practice on word problems and reverse calculations.

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