Formula for `n^(th)` term of a sequence - consistent difference

In order to predict the `n^(th)` term of a sequenece you will need to create a formula.

For sequence patterns of consistent differences finding the `n^(th)` term of a linear sequence (an addition or subtraction sequence) is worked out by:

Use the following mnemonic:

formula of a sequence when numbers are consistently different

Common different x `n` + term before first term

or in more detail

1. Find the consistent difference and multiply by `n` (`n` = term position)
2. Find the value before the first term (what would the value be if it were there)
3. Add the two together

Summary

Full value of the `n`^th = common different x `n` + term before first term.

Or

Formula for `n`^th term = difference x `n` + (first term — difference).

Example 1

Even numbers

What is the formula for the `n`^th term for the following sequence, and what is the `11^(th)` term?

`2`, `4`, `6`, `8`, `10`, `12`

To tackle this always find the sequence pattern.

First, see if there is a consistent difference between each number.

Extend this sequence to its 11th term, the difference will still be 2

Yes and that number is `2`

Now to find the formula:

i. Find the difference and multiply by `n`

The difference is `2`

Which we now multiply by `n`

Therefore this equals `2n`

ii. Find the value before the first term.

First term	`=`	`2`
Consistent difference	`=`	`2`
Take them away	`=`	`0`

The value before the first term `=0`

iii. Adding the two together we get the formula for this sequence as:

Formula for `n^(th)` term `=2n+0`

Answer: The formula for the `n^(th)` term `=2n`

NOTE:

This formula is well known as the sequence progression for even numbers.

Always check the formula is correct.

Try different values of `n` in the formula `2n`

If `n=1` therefore term `=2times1=2`

If `n=2` therefore term `=2times2=4`

If `n=3` therefore term `=2times3=6`

This is correct

Now we have the correct formula we can work out the other terms.

The `11^(th)` term would be:

If `n=11` then `2n=2times11=22`

Answer: The `11^(th)` term `=22`

Example 2

Odd numbers

What is the formula for the `n^(th)` term for the following sequence, and what is the `14^(th)` term.

`1`, `3`, `5`, `7`, `9`, `11`

To tackle this always find the sequence pattern.

First, see if there is a consistent difference between each number.

Extend this sequence to its 14th term, the difference will still be 2

Yes and that number is `2`

Now to find the formula:

i. Find the difference and multiply by `n`

The difference is `2`

Which we now multiply by `n`

Therefore this equals `2n`

ii. Find the value between the first term and the consistent difference.

First term	`=`	`1`
Consistent difference	`=`	`2`
Take them away	`=`	`-1`

The value before the first term `=-1`

iii. Adding the two together we get the formula for this sequence as:

Formula for `n^(th)` term `=2n-1`

Answer: The formula for the `n^(th)` term `=2n-1`

NOTE:

This formula is well known as the sequence progression for odd numbers.

Always check the formula is correct.

Try different values of `n` in the formula `2n-1`

If `n=1` therefore term `=2times1-1=1`

If `n=2` therefore term `=2times2-1=3`

If `n=3` therefore term `=2times3-1=5`

If `n=4` therefore term `=2times4-1=7`

This is correct

Now we have the correct formula we can work out other terms.

The `14^(th)` term would be:

If `n=14` then `2n-1=2times14-1=27`

Answer: The `14^(th)` term `=27`

Example 3

What is the formula for the `n^(th)` term for the following sequence, and what is the `55^(th)` term.

`6`, `8`, `10`, `12`, `14`

To tackle this always find the sequence pattern.

First, see if there is a consistent difference between each number.

Extend this sequence to its 55th term, the difference will still be 2

Yes and that number is `2`

Now to find the formula:

i. Find the difference and multiply by `n`

The difference is `2`

Which we now multiply by `n`

Therefore this equals `2n`

ii. Find the value between the first term and the consistent difference.

First term	`=`	`6`
Consistent difference	`=`	`2`
Take them away	`=`	`4`

The value before the first term `=4`

iii. Adding the two together we get the formula for this sequence as:

Formula for `n^(th)` term `=2n+4`

Answer: The formula for the `n^(th)` term `=2n+4`

Always check the formula is correct.

Try different values of `n` in the formula `2n+4`

If `n=1` therefore term `=2times1+4=6`

If `n=2` therefore term `=2times2+4=8`

If `n=3` therefore term `=2times3+4=10`

This is correct

Now we have the correct formula we can work out the other terms.

The `55^(th)` term would be:

If `n=55` then `2n+4=2times55+4=114`

Answer: The `55^(th)` term `=114`

Example 4

What is the formula for the `n^(th)` term for the following sequence, and what is the `7^(th)` term.

`12`, `10`, `8`, `6`, `4`

To tackle this always find the sequence pattern.

First, see if there is a consistent difference between each number.

Extend this sequence to its 7th term, the difference will still be minus 2

Yes and that number is `-2`

Now to find the formula:

i. Find the difference and multiply by `n`

The difference is `-2`

Which we now multiply by `n`

Therefore this equals `-2n`

ii. Find the value between the first term and the consistent difference.

First term	`=`	`12`
Consistent difference	`=`	`-2`
Take them away	`=`	`14`

The value before the first term `=14`

iii. Adding the two together we get the formula for this sequence as:

Formula for `n^(th)` term `=-2n+14`

Answer: The formula for the `n^(th)` term `=-2n+14`

Always check the formula is correct.

Try different values of `n` in the formula `-2n+14`

If `n=1` therefore term `=-2times1+14=12`

If `n=2` therefore term `=-2times2+14=10`

If `n=3` therefore term `=-2times3+14=8`

This is correct

Now we have the correct formula we can work out other terms.

The `7^(th)` term would be:

If `n=7` then `-2n+14=-2times7+14=0`

Answer: The `7^(th)` term `=0`