Generate a Sequence from an Nth Term

To generate a sequence from the nth term, we have to remember what an nth term actually means.

It is a position-to-term rule which allows the position within the sequence to be inputted to generate the value which corresponds to this position in the sequence.

Let's look at this in practice now to learn how we should approach finding a sequence from the nth term.

e.g. Find the first 5 terms in this sequence: 4n - 6

For this, we have to substitute the values n = 1, n = 2, n = 3, n = 4 and n = 5 into the nth term:

When n = 1, 4n - 6 = (4 × 1) - 6 = -2

When n = 2, 4n - 6 = (4 × 2) - 6 = 2

When n = 3, 4n - 6 = (4 × 3) - 6 = 6

When n = 4, 4n - 6 = (4 × 4) - 6 = 10

When n = 5, 4n - 6 = (4 × 5) - 6 = 14

So our first five terms for the sequence 4n - 6 are: -2, 2, 6, 10, 14.

Let's look at another example now which asks us to apply our knowledge of the nth term to solve a problem. 

e.g. Find the difference between the 100^th and the 90^th terms in this sequence: 5n + 2

The approach to this is very similar to the last question, but we use n = 90 and n = 100 instead:

When n = 100, 5n + 2 = (5 × 100) +2 = 502

When n = 90, 5n + 2 = (5 × 90) +2 = 452

We can then find the difference between these two terms by subtracting these two values:

502 - 450 = 50

So the difference between the 90^th and 100^th terms in the sequence 5n + 2 is 50. 

In this activity, we will use nth terms to find specified values in a sequence and to solve problems involving the relationships between terms in a sequence.