Arithmetic Explicit Formula

Now that we know the starting value, a1, of arithmetic sequences and the common difference, d, we can create an explicit formula to find the value of any term in the sequence. When looking at the table below:

we can see that we know the value at positions 1, 2, 3, and 4. They are 2, 4, 6, and 8 respectively. However, what if we wanted to know the value of the 100th term. We would have to add the common difference another 96 times. That is a lot of button pushing, therefore we can construct an Explicit Formula instead.

The explicit formula for arithmetic sequences is as follows:

an = a1 + (n-1) * d

we know that a1 is the starting value and d is the common difference.

Remember that if your common difference is subtraction, use a negative d value.

an is the value of the term we are looking for and n is the term we are looking for.

So, for our table above the a1 = 2 and d = 2 so our equation is:

an = 2 + (n-1) * 2

Now in order to find the 100th term we have to plug in 100 for n and write an as a100.

a100 = 2 + (100 - 1) * 2

an = 2 + (99) * 2

a100 = 2 + 198

a100 = 200

We can see that the value at term 100 is 200.

What if we wanted to know when our sequence reached 80. Here we would plug 80 in for an

80 = 2 + (n-1) * 2

First, we can subtract 2 from both sides

78 = (n-1) * 2

Then, we can divide by 2

39 = n - 1

Finally, we add 1 to both sides

40 = n

so our equation is equal to 80 at position 40.