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Arithmetic sequences calculator

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Arithmetic sequences calculator

This online tool can help you find n^th term and the sum of the first n terms of an arithmetic progression. Also, this calculator can be used to solve much more complicated problems. For example, the calculator can find the common difference (d) if a₅ = 19 and S₇ = 105. The main advantage of this calculator is that it will generate all the work with detailed explanation.

Tutorial

What is an arithmetic sequence?

An arithmetic progression is a list of numbers in which the difference between any two successive members is a constant.

Example 1: The sequence of numbers 2, 5, 7, 10, 13,... is arithmetic because the difference between any two consecutive numbers is equal to 3.

Example 2: The sequence 2, 3, 5, 8,... is not arithmetic because the difference between terms is not constant.

Arithmetic sequence notation

a₁

the first term of a sequence

common difference

a_n

n - th term of a sequence

S_n

sum of first n terms

Example 3: In the previous sequence, 2, 5, 7, 10, 13,... we have a1 = 2, d = 3, a₃= 7, S₄ = 24.

Arithmetic sequence formulas

d = a₂ - a₁

a_n = a₁ + (n-1)d

S_n = n/2 [ 2a₁ + (n-1)d]

S_n = n/2 [ a₁ + a_n ]

Worked examples

Example 4: Find the 7-th element of an arithmetic sequence with d = 3 and a₁ = 5.

a_n = a₁ + (n-1)d
a₇ = 5 + (7-1)3
a₇ = 5 + 6·3
a₇ = 5 + 18
a₇ = 23

Example 5: Find the sum of first 10 elements of a sequence { 5, 9, 13, 17, 21,...}.

In this example we have a1 = 5, d = 4, and n = 10 since we want to sum up the first n terms.

S_n = n/2[2a1+(n+1)d]
a₁₀ = 10/2[2*5+(10+1)4]
a₁₀ = 5[10+11*4]
a₁₀ = 5[10+44]
a₁₀ = 5[54]
a₁₀ = 108

Basic calculator Solve for a₁ and d Solve for n

Arithmetic Sequences Calculator

Find n - th term and the sum of the first n terms.
help ↓↓ examples ↓↓

First term (a₁) :

Common difference (d) :

Number of Terms (n) :

Choose what to compute

a_n (the n-th term)

S_n (the sum of the first n terms )

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Arithmetic Sequences Calculator

Use this one if you need to find either a1 or d.
help ↓↓ examples ↓↓

Enter values in two out of four rows.

First term (a₁):

Common difference (d):

Enter n-th term

a( )=

Sum of first n terms

S( )=

Arithmetic Sequences Calculator

This calc will find unknown number of terms.
help ↓↓ examples ↓↓

Enter values in three out of four rows.

First term (a₁) :

Common difference (d) :

Enter n-th term

a(n) =

Sum of first n-terms

S(n) =

Examples

ex 1:

1 + 2 + 3 + 4 + . . . + 98 + 99 + 100 = ?

ex 2:

The first term of an arithmetic sequence is equal to 5/2 and the common difference is equal to 2. Find the value of the 20^th term

ex 3:

An arithmetic sequence has a common difference equal to 7 and its 8^th term is equal to 43. Find its first term.?

ex 5:

The first term of an arithmetic progression is -12, and the common difference is 3 determine how many terms must be added together to give a sum of 1104.?

Find more worked-out examples in our database of solved problems.

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RESOURCES

1. Sum of an arithmetic progression - formula derivation

2. Arithmetic progression formulas

3. Video, Quiz and Course

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