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Nth Term - Finding and verifying a formula for the nth term of a sequence.

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Introduction

Nth Term - Finding and verifying a formula for the nth term of a sequence.

I shall give some examples of how to go about it.

If the numbers in the sequence increase in EQUAL STEPS then things are fairly straightforward. For example:

5 , 8 , 11 , 14 , 17 , ... (step length 3)
26 , 31 , 36 , 41 , 46 , ...
(step length 5)
20 , 18 , 16 , 14 , 12 , ...
(step length -2)

First you, draw up a table, giving each term its "counter" (generally called n)

 n 1 2 3 4 5 . . . Term 5 8 11 14 17 . . .

The common step length is 3. So the formula will be
3×n   +   something (You can complete this something later on)

 This is because, if the step length is the same for all the terms in the sequence, the formula will be of the format:step × n   +   something

For the sequence above, the rule 3×n   +   something

Middle

2

3

4

5

. . .

Term

26

31

36

41

46

. . .

The common step length is 5. So the formula will be
5×n   +   something

For the sequence above, the rule 5×n   +   something would give the values
5×1 +
something = 5 + something
5×2 +
something = 10 + something
5×3 +
something = 15 + something
5×4 +
something = 20 + something
5×5 +
something = 25 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the
something is +21
So the formula for the nth term is   5n + 21

It’s a simple Matter of Comparing every sequence you get, as this is vital to help you find the “something”

This time we just had a higher difference in the sequence.

Now let's do the third example....

 n 1 2 3 4 5 . . . Term 20 18 16 14 12 . . .

The common step length is -2.

Conclusion

If the terms do NOT increase in equal steps, then you have to think about things a bit more. Here is an example:

4 , 7 , 12 , 19 , 28 , . . . . steps are +3, +5, +7, +9

 If the steps between the terms are not equal, you should try a rule based onn2   + something
 n 1 2 3 4 5 . . . Term 4 7 12 19 28 . . .

For the sequence above, the rule n2   +   something would give the values
1
2 + something = 1 + something
2
2 + something = 4 + something
3
2 + something = 9 + something
4
2 + something = 16 + something
5
2 + something = 25 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the
something is +3
So the formula for the nth term is   n2 + 3

This is a topic where practice will help you to see the patterns and rules more easily and I have definitely learnt that.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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