Arithmetic Sequence Calculator

Definie Arithmetic Sequence

A sequence is a list of numbers in a particular order. Each number in a sequence is called a term. The first term is represented by a₁, the second term is represented by a₂ , and so on. a_n represents the nth term of the sequence. One kind of sequence is an arithmetic sequence. In an arithmetic sequence each term after the first is found by adding a constant, called the common difference, d, to the previous term.

A simple way to generate a sequence is to start with a number a, and add to it a fixed constant d, over and over again. This type of sequence is called an arithmetic sequence.

An arithmetic sequence is a sequence of the form: a, a + d, a + 2d, a + 3d, a + 4d, ........
The number a is the first term, and d is the common difference of the sequence.

nth term of arithmetic sequence

The nth term of an arithmetic sequence is given by: a_n = a₁ + (n – 1)d

The number d is called the common difference because any two consecutive terms of an arithmetic sequence differ by d, and it is found by subtracting any pair of terms a_n and a_n+1.
That is: d = a_n+1 – a_n

Sum of arithmetic sequence

To find a formula for the sum, S_n, of the first n terms of an arithmetic sequence, we can write out the terms as
S_n = a + (a+d) + (a+2d) +......+[a+(n-1)d]

The final equation of Sum of arithmetic sequence is: S_n = n/2(a + a_n)

Example for finding the terms of an Arithmetic Sequence

Example: Find the nth term, the fifth term, and the 100th term, of the arithmetic sequence determined by a = 2 and d = 3.

Solution: To find a specific term of an arithmetic sequence, we use the formula for finding the nth term. The nth term of an arithmetic sequence is given by an = a + (n – 1)d.

So, to find the nth term, substitute the given values a = 2 and d = 3 into the formula.

an = 2 + (n – 1)3

Now, to find the fifth term, substitute n = 5 into the equation for the nth term.

a₅ = 2 + (5 – 1)3 = 14

Finally, find the 100th term in the same way as the fifth term.

a₁₀₀ = 2 + (100 – 1)3 = 299