# The nth Term

## The nth Term

The 'nth' term is a formula with 'n' in it which enables you to find **any** term of a sequence without having to go up from one term to the next.

'* n*' stands for the

*so to find the 50th term we would just substitute 50 in the formula in place of '*

**term number***'.*

**n****There are two types of sequences that you will have to deal with:**

This is when the difference between terms is always the same.

e.g. 1, 4, 7, 10, ... This has a difference which is always 3.

**How do you find the formula for the 'nth' term?**

Well, the three times table has the formula '3n' and the terms in this sequence are two less than the terms in the three times table so the formula is '3n - 2'.

You can always find the 'nth term' by using this formula:

**nth term = dn + (a - d)**

Where **d** is the difference between the terms, **a **is the first term and **n** is the term number.

e.g. 6, 11, 16, 21, ...For this sequence **d = 5, a = 6**

So the formula is **nth term = 5n + (6 - 5)**

which becomes **nth term = 5n + 1**

**What if the difference keeps changing?**

Obviously these are more difficult but once again we can use a formula!

**nth term = a + (n - 1)d + ½(n - 1)(n - 2)c**

This time there is a letter * c* which stands for the

**second difference**(or the difference between the differences and

*is just the difference between the*

**d****first two**numbers.

Putting the right numbers into the formula is reasonably simple (once you've learnt the formula!). Simplifying it requires good Algebra skills so **practice your Algebra!**

**Here's an example:**

Here the difference between the first two numbers is **1** so **d** = **1**

Also the **second differences are 2** so **c** = **2** The first term is **2** so **a** = **2**

Using the formula, **nth term = 2 + (n - 1)x1 + ½(n - 1)(n - 2)x2**

Getting rid of brackets (and noticing that ½ x 2 = 1):

**nth term = 2 + n - 1 + n2 - 3n + 2**

Simplifying,

**nth term = n ^{2} - 2n + 3**

If all that gave you a headache there is an alternative way!

**1.** If the first differences keep changing but the second difference is constant then the formula is something to do with '**n ^{2}**'. Make a table showing the first few terms of '

**n**'.

^{2}**2.** In the next column of your table write the differences between the term of '**n ^{2}**' and your sequence.

**3.** Find the formula for **this** new sequence using **dn + (a - d)**

**4.** Add it on to **n ^{2}** to give yo your final formula.

**Have a look at this using the sequence above:**

Sequence | 'n^{2}' | Difference |

2 | 1 | 1 |

3 | 4 | -1 |

6 | 9 | -3 |

11 | 16 | -5 |

18 | 25 | -7 |

**For the sequence 1, -1, -3, -5, -7 => a = 1 and d = -2**

So the formula is * -2n + (1 - -2) *which simplifies to

**-2n + 3**Final Formula (Step 4) : * nth term = n^{2} -2n + 3 *(as we got before!)