N^th term of an arithmetic or geometric sequence

Tutorial

About this calculator?

The primary purpose of this calculator is to analyze sequences. If we are given the first few terms of a sequence, it will find the general term as well as the next few members.

Examples

Example 01:

Find the next element of a sequence.

2, 5, 8, 11, 14, ?

Solution

First, we'll identify the sequence. To do so, we need to calculate the difference between consecutive terms.

5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3

The difference is always the same, so we can conclude that the series is arithmetic. The first element is a₁ = 2 and the difference is d = 3. The missing element of a sequence is

? = 14 + 3 = 17

The general term of a sequence is

a_n = a₁ + (n-1)d
a_n = 2 + 3*(n-1)
a_n = 2 + 3 * n- 3
a_n = -1 + 3 n

Example 02:

Find the next element of a sequence.

2, 6, 18, 54, ?

Solution

In this example the sequence is not arithmetic (the differences are not constant), so we'll try to show that this sequence is geometric. To do so, we will compute the quotient of any two consecutive terms.

6:2 = 3, 18:6 = 3, 54:18 = 3

Since the quotients are always the same, we can conclude that the series is geometric. The first element is b₁ = 2 and the common ratio is r = 3. The missing element of a sequence is

? = 54 * 3 = 162

The general term of a sequence is

b_n = b₁*r^n-1
a_n = 2 * 3^n-1

Example 03:

Identify sequence

1, 3, 9, 19, 33, 51

Solution

We can easily see that this is not an arithmetic or geometric series, so we need to find the second differences to prove that the sequence is quadratic.

The first differences are:

3 - 1 = 2
9 - 3 = 6
19 - 9 = 10
33 - 19 = 14
51 - 33 = 18

The second differences are:

6 - 2 = 4
10 - 6 = 4
14 - 10 = 4
18 - 14 = 4

We can see that the second differences are constant, meaning that the series is quadratic.

The general term is:

c_n = 2n² - 4n + 3