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)

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

For the sequence above, the rule 3×n + something would give the values
3×1 + something = 3 + something
3×2 + something = 6 + something
3×3 + something = 9 + something
3×4 + something = 12 + something
3×5 + something = 15 + something
You then compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +2
So the formula for the nth term is 3n + 2 That’s Easy Enough!

A ...

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A second example....

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....

The common step length is -2. So the formula will be
-2×n + something

For the sequence above, the rule -2×n + something would give the values
-2×1 + something = -2 + something
-2×2 + something = -4 + something
-2×3 + something = -6 + something
-2×4 + something = -8 + something
-2×5 + something = -10 + something
Compare these values with the ones in the actual sequence - it should be obvious that the value of the something is +22
So the formula for the nth term is -2n + 22 or written more neatly 22 - 2n

Even though this looks hard thanks to the negative number there is no difference the rule stays the same.

Now to go on to a harder version of the rule but it’s pretty easy to get the grasp of, I say this because I got the grasp of it easy and if I can anyone can!

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

For the sequence above, the rule n2 + something would give the values
12 + something = 1 + something
22 + something = 4 + something
32 + something = 9 + something
42 + something = 16 + something
52 + 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