Find the Formula for the General Term of a Sequence

In this activity, we will learn how to find the formula for the general term T_n of a sequence.

Example

Term number n =                   1           2            3            4            5
Term, T_n=                               4           8           12          16          20

The common difference is +4

The common difference between the terms is +4, so we write this as 4n.

It is clear, that T_n = 4n.

Compare this sequence with the one below:

Term number n =                   1            2            3            4            5
Term, T_n=                               7           11          15          19          23

The common difference is +4

The common difference of +4 gives us 4n again.

But this time, we need to do something more. 

If we test out 4n with the first term in the table, we get 4 x 1 = 4 but the first term isn't 4, it is actually 7!

This means that we need to do an adjustment to the formula: 7 - 4 = 3, so we know that we need to add 3 to the answer of 4n.

This gives us the corrected formula of 4n + 3

To finish off, we need to test that formula out with some of the other terms to make sure it works:

n = 2     4 x 2 = 8 + 3 = 11, which is correct.

          n = 3     4 x 3 = 12 + 3 = 15, which is also correct!

So T_n = 4n + 3

It's a bit tricky, but we'll take it step by step and work through the questions carefully.

Let's get started!