How to Solve Arithmetic Sequences? (+FREE Worksheet!)
Do you want to know how to solve Arithmetic Sequences problems? you can do it in few simple and easy steps.
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Step by step guide to solve Arithmetic Sequences problems
- A sequence of numbers such that the difference between the consecutive terms is constant is called arithmetic sequence. For example, the sequence \(6, 8, 10, 12, 14\), … is an arithmetic sequence with common difference of \(2\).
- To find any term in an arithmetic sequence use this formula: \(\color{blue}{x_{n}=a+d(n-1)}\)
- \(a =\) the first term,\(d =\) the common difference between terms, \(n =\) number of items
Arithmetic Sequences – Example 1:
Find the first three terms of the sequence. \(a_{17}=38,d=3\)
Solution:
First, we need to find \(a_{1}\) or a. Use arithmetic sequence formula: \(\color{blue}{x_{n}=a+d(n-1)}\)
If \(a_{8}=38\), then \(n=8\). Rewrite the formula and put the values provided:
\(x_{n}=a+d(n-1)→38=a+3(3-1)=a+6\), now solve for \(a\).
\(38=a+6→a=38-6=32\),
First three terms: \(32,35,38\)
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Arithmetic Sequences – Example 2:
Given the first term and the common difference of an arithmetic sequence find the first five terms. \(a_{1}=18,d=2\)
Solution:
Use arithmetic sequence formula: \(\color{blue}{x_{n}=a+d(n-1)}\)
If \(n=1\) then: \(x_{1}=18+2(1)→x_{1}=18\)
First five terms: \(18,20,22,24,26\)
Arithmetic Sequences – Example 3:
Given the first term and the common difference of an arithmetic sequence find the first five terms. \(a_{1}=24,d=2\)
Solution:
Use arithmetic sequence formula: \(\color{blue}{x_{n}=a+d(n-1)}\)
If \(n=1\) then: \(x_{1}=22+2(1)→x_{1}=24\)
First five terms: \(24,26,28,30,32\)
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Arithmetic Sequences – Example 4:
Find the first five terms of the sequence. \(a_{17}=152,d=4\)
Solution:
First, we need to find \(a_{1}\) or \(a\). Use arithmetic sequence formula: \(\color{blue}{x_{n}=a+d(n-1)}\)
If \(a_{17}=152\), then \(n=17\). Rewrite the formula and put the values provided:
\(x_{n}=a+d(n-1)→152=a+4(17-1)=a+64\), now solve for \(a\).
\(152=a+64→a=152-64=88\),
First five terms: \(88,92,96,100,104\)
Exercises
Given the first term and the common difference of an arithmetic sequence find the first five terms and the explicit formula.
- \(\color{blue}{a_{1} = 24, d = 2}\)
- \(\color{blue}{a_{1} = –15, d = – 5}\)
- \(\color{blue}{a_{1} = 18, d = 10}\)
- \(\color{blue}{a_{1 }= –38, d = –100}\)
Download Arithmetic Sequences Worksheet
- First Five Terms \(\color{blue}{: 24, 26, 28, 30, 32, Explicit: a_{n} = 22 + 2n}\)
- First Five Terms \(\color{blue}{: –15, –20, –25, –30, –35, Explicit: a_{n} = –10 – 5n}\)
- First Five Terms \(\color{blue}{: 18, 28, 38, 48, 58, Explicit: a_{n} = 8 + 10n}\)
- First Five Terms \(\color{blue}{: –38, –138, –238, –338, –438, Explicit: a_{n} = 62 – 100n}\)
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