### Sequences

#### Jumping to Later Terms of APs

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# Learning Objectives

######

###### Jumping to Later Terms of APs

If we used the same thought process that we use for simple sequences, we would recall that sequences follow the below progression

*a, a+d, a+2d, a+3d, a+4d, a+5d, …*

Note that for T₂ the common difference, d had been added once, for T₃ it had been added twice, for T₄ it has been added three times, and so on.

Knowing this, we can conclude that** arithmetic progressions have an ****n****th term given by the rule**

** **

**Tₙ = a + (n-1)d**

**Tₙ = a + (n-1)d**

Where

a = first term; and

d = common difference.

**This format of the rule is known as the ****n****th term rule****.**

###### Example

Suppose that an AP has a 45th term of 180 and a 57th term of 228. Find

The 62nd term; and

The 1st term.

**Solution:**

Firstly, create a pair of simultaneous equations using the information given.

*180 = a + 44d*

* *

*228 = a + 56d*

* *

Now, solve the equations to find the value of d.

*-48 = -12d*

* *

*d = 4*

* *

Substitute the value of d into one of the original equations to find the value of a.

*180 = a + 44(4)*

* *

*a = 4.*

* *

We should now use this information to find solve the question.

Find the 62nd term.

T₆₂ = 4 + (61)(4) = 248.

2. Find the first term.

T₁ = 4.