What is Arithmetic Sequence

Read,5 minutes

Arithmetic Sequences Definition

A list of numbers with a clear pattern is known as an arithmetic sequence. We can tell if a series of numbers is arithmetic by taking any number and subtracting it by the number before it. In an arithmetic sequence, the difference between two numbers stays the same.

The letter \(d\) stands for the common difference, the constant difference between the numbers in an arithmetic sequence. We have an increasing arithmetic sequence if \(d\) is positive. We have a decreasing arithmetic sequence if \(d\) is negative.

Arithmetic Sequences Formula

We can use the following formula to find different terms for the arithmetic sequence:

\(a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)\), where:

\(a_1\): The first term
\(a_n\): The \(n\)th term
\(n\): The term poistion
\(d\): Common difference

How to figure out the terms in an arithmetic sequence?

To find any term in the arithmetic sequence, we need to know a common difference, the position of the term we want to find, and a term in the sequence.

Example

Consider the arithmetic sequence: \(3, \ 7, \ 11, \ 15, \ 19, \ ...\), Find \(a_{28} \ = \ ?\)

Solution

Before using the arithmetic sequence formula, we need to know the first term, the common difference, and the position of the term we want to find:

The first term: \(a_1 \ = \ 3\)
common difference: \(d \ = \ 4\)
Term's position: \(n \ = \ 28\)

Now, we put these numbers into the formula and find the answer:

\(a_n \ = \ a_1 \ + \ d \ (n \ –\ 1) \ ⇒ \ a_{28} \ = \ 3 \ + \ 4(28 \ - \ 1) \ ⇒ \ a_{28} \ = \ 111\)

Summing an Arithmetic Series

To summing the terms in an arithmetic sequence, use this formula:

\(\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} (2a \ + \ (n \ - \ 1)d)\)

Example

Find the sum of the first \(12\) terms for the sequence: \(1, \ 6, \ 11, \ 16, \ 21, \ ...\)

Solution

The first term: \(a_1 \ = \ 1\)
common difference: \(d \ = \ 5\)
\(n \ = \ 12\)

So:

\(\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} (2a \ + \ (n \ - \ 1)d) \ ⇒\)

\(\sum_{k \ = \ 0}^{11} \ (1 \ + \ 5k) \ = \ \frac{12}{2} (2 \ + \ (11)5) \ = \ 6 \times (57) \ = \ 342\)

Arithmetic Sequences

Think of this lesson as more than a rule to memorize. Arithmetic Sequences is about patterns, terms, common differences, ratios, and sums. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

For arithmetic sequences, use \(a_n=a_1+(n-1)d\). For geometric sequences, use \(a_n=a_1r^{n-1}\). The pattern tells you which formula fits.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Decide whether the pattern adds or multiplies.
Identify the first term and the difference or ratio.
Choose the term or sum formula.
Check whether the answer makes sense in the pattern.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Arithmetic Sequences

1) Find the explicit formula: \(4, \ 9, \ 14, \ 19, \ ...\)

2) Find the explicit formula: \(-3, \ 1, \ 5, \ 9, \ ...\)

3) Find \(a_{15}\): \(12, \ 7, \ 2, \ -3, \ ...\)

4) Find \(a_{20}\) if \(a_1 \ = \ 8\) and \(d \ = \ 3\)

5) Find the explicit formula if \(a_7 \ = \ 31\) and \(d \ = \ 4\)

6) Find \(a_{25}\) if \(a_{10} \ = \ -12\) and \(d \ = \ -3\)

7) Find \(a_{12}\): \(6, \ 10.5, \ 15, \ 19.5, \ ...\)

8) Find the explicit formula if \(a_5 \ = \ 18\) and \(a_{12} \ = \ 53\)

9) Insert four arithmetic means between \(3\) and \(28\)

10) Find the sum of the first \(20\) terms of \(7, \ 11, \ 15, \ ...\)

11) Find \(n\) if \(59\) is a term of \(5, \ 8, \ 11, \ ...\)

12) Given \(a_3 \ = \ 11\) and \(a_{18} \ = \ 56\), find \(a_{30}\)

13) Find \(S_{15}\) for \(-10, \ -6, \ -2, \ ...\)

14) An auditorium has \(25\) rows. The first row has \(18\) seats, and each next row has \(2\) more seats. How many seats are there?

15) Find \(a_{25}\) if \(a_4 \ = \ 21\) and \(a_{16} \ = \ -15\)

16) Find the sum of the multiples of \(7\) from \(14\) through \(140\)

17) If \(a_{12} \ = \ 50\) and \(a_{28} \ = \ 130\), find \(a_{20}\)

18) If \(a_1 \ = \ -8\), \(d \ = \ 5\), and \(a_n \ = \ 82\), find \(n\) and \(S_n\)

19) Find \(n\) if \(a_1 \ = \ 4\), \(d \ = \ 6\), and \(S_n \ = \ 444\)

20) An arithmetic sequence has \(a_6 \ = \ 17\) and \(S_{20} \ = \ 730\). Find \(a_1\) and \(d\)

Answers

1)Find the explicit formula: \(4, \ 9, \ 14, \ 19, \ ...\)