Question

State the recursive definition (include the initial conditions) for each of the following sequences. (a) (1, 2, 4, 8, 16, 32, 64,...) (b) (0, 2, 6, 12, 20, 30, 42, 56, ...) (c) (4, 5, 7, 11, 19, 35, 67, 131, 259,...)

Answer 1

I'll focus on part (a) only.

The first term is 1, and we double each term to get the next term.

• 1*2 = 2
• 2*2 = 4
• 4*2 = 8
• 8*2 = 16
• 16*2 = 32

And so on. The notation
`a_1 = 1` means "the first term is 1". The subscript 1 tells us which term number we're at. Term
`a_2` is the second term, then
`a_3` is the third term, etc.

The recursive step could be written as
`a_n = 2*a_(n-1)` to mean "the nth term is double of the previous term".

We write the final answer as

`\begin{cases}a_1 = 1\\a_n = 2*a_(n-1)\end{cases}`

The first line is the initial condition. The second line is the recursive step to generate each successive term. If we wanted say the 100th term, then we'd have to generate the previous 99 terms, which would be a lot of tedious work to do by hand.