Question

How do you find the explicit formula for the following sequence 10, 14, 18, 22, 24, ...?

Answer 1

Hi there! As mentioned in the comments below, I believe that the 24 should be a 26 to make this a nice sequence. If this is true, this is an arithmetic sequence with a common difference of 4.

Explanation:

The expression for the explicit formula of an arithmetic sequence is:

`x_n = a + d(n-1)`

Where:

`x_n` = the "n-th" term in the sequence

`a` = the first term in the sequence

`d` = the common difference between the terms in the sequence

`n` = the term number (index)

Looking at the sequence:

10, 14 18, 22, 26 we can deduce that:

`a = 10`
`d = 4`

And this is all you need, since you're finding an explicit formula (a formula that works for any n value).

Substituting this information into the general formula we get:

`x_n = 10 + 4(n-1)`

If you want, you can simplify this:

`x_n = 10 + 4n-4`

`x_n = 6 + 4n`

And that's it! That's the explicit formula! After determining the general formula for a sequence, it is always a good idea to double-check to see if it actually works with the numbers in your sequence. Lets do that:

We know, by extension, that the 6th term in the sequence should be 30 and therefore, if we sub 6 into "n" of our general formula we get:

`x_n = 6 + 4n`

`x_6 = 6 + 4(6)`

`x_6 = 6 + 24`

`x_6 = 30`

We got what we were looking for! Thus, our explicit formula works! Hopefully this was all clear and straightforward! If you have any questions, please feel free to let me know! :)