Arithmetic Sequences

Arithmetic Sequences

Do you want to know how to solve Arithmetic Sequences problems? you can do it in few simple and easy steps.

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

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: \(32,35,38\)

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: \(18,20,22,24,26\)

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: \(24,26,28,30,32\)

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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