# How to Solve 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

### 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: $$32,35,38$$

### 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: $$24,26,28,30,32$$

### 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.

• $$\color{blue}{a_{1} = 24, d = 2}$$
• $$\color{blue}{a_{1} = –15, d = – 5}$$
• $$\color{blue}{a_{1} = 18, d = 10}$$
• $$\color{blue}{a_{1 }= –38, d = –100}$$

• 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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