Question

A recursive rule for a sequence is given. Find the first four terms of the sequence.

Answer 1

Answer:  

So...

   a0 = -2

   a1 = (a0)2 - 4  =  4 - 4  =  0

   a2 = (a1)2 - 4 = ((a0)2-4)2 - 4 =...=  02 - 4  = -4  

       For an+1, you may use the previous term (an) if you have just calculated it, rather than calculating it recursively again

   a3 = (-4)2 - 4 = 12

 

The first four terms in the sequence are -2, 0, -4, 12.

Step-by-step explanation:

This recursive sequence is defined as follows:

 a0 = -2      [the first term]

 an+1 = an2 - 4     [for all other terms, n≥0; note that you cannot find a0 using this line]

 

Note that this is not "BASE" for a number base, but it is "SUB" for subscript, indicating the term.  Sometimes, sequences are written with the first term being a1 and sometimes sequences are written with the first term being a0.  This is because there are situations that make one or the other more convenient (for example, starting with elapsed time t=0 makes sense).

 

To find the value of later terms, a recursive definition requires that you find the value of the previous term, which requires that you find the value of the previous term, which requires that you find the value of the previous term, which requires that you find the value of the previous term, ... which requires that use the value of the first term.

 

That's why (joke) that the dictionary definition of "recursive" is "see recursive."