# Expression and Explicit Formula

Rather than write a recursive formula, we can write an explicit formula. The explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use FVDFDFHCSDBCKISDABHCIOSDCKSDNCOISDHCNSDICHSDOICNSDLCIJHSDIOV- CSDKNJCOISDHNVDSVNIOSDVNHDISLVNHSDIOVHSFDNVNHISDVIDSNVSDLKVN HSDVSDNCOSDNCKSDLCHIOSDCNSDOICHNOIDSCSAOICASDICSDOCDSIOCSDKC- HDIVCNDSCISDCIDSCJJCJDOSOJSJDFOJSJFSJJFEEThis lesson will work with arithmetic sequences, their recursive and explicit formulas and finding terms in a sequence. In this lesson, it is assumed that you know what an arithmetic sequence is and can find a common difference. If you need to review these topics, click here.

Let’s look at the arithmetic sequence

20, 24, 28, 32, 36, . . .

This arithmetic sequence has a common difference of 4, meaning that we add 4 to a term in order to get the next term in the sequence.

The recursive formula for an arithmetic sequence is written in the form

For our particular sequence, since the common difference (d) is 4, we would write

So once you know the common difference in an arithmetic sequence you can write the recursive form for that sequence.

However, the recursive formula can become difficult to work with if we want to find the 50th term. Using the recursive formula, we would have to know the first 49 terms in order to find the 50th. This sounds like a lot of work. There must be an easier way. And there is!

Rather than write a recursive formula, we can write an explicit formula. The explicit formula is also sometimes called the closed form. To write the explicit or closed form of an arithmetic sequence, we use...

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