Question

For each of the following sequences, • Give a formula for the nth term in the sequence, • Give a recursive definition for the sequence (ie. initial values and a recursive equation). (a) 1, 2, 3, 4, 5, . . . (b) 1, 2, 4, 8, 16, 32, . . . (c) 1, 2, 6, 24, 120, . . .

Answer 1

(a)
`n^(th) = n`

`f(1) = 1`

`f(n) = f(n-1) + 1`

(b)
`n^(th) = 2^(n-1)`

`f(1) = 1`

`f(n) = f(n-1) * 2`

(c)
`n^(th) = n!`

`f(1) = 1`

`f(n) = f(n-1) * n`

Explanation:

(a) This is a sequence of consecutive number

`n^(th) = n`

`f(1) = 1`

`f(n) = f(n-1) + 1`

(b) This is a sequence of 2 to the power of n - 1. The next number is twice time of this number

`n^(th) = 2^(n-1)`

`f(1) = 1`

`f(n) = f(n-1) * 2`

(c) This is factorial sequence. Where the next number is this number multiplied by
`n^(th)`

`n^(th) = n!`

`f(1) = 1`

`f(n) = f(n-1) * n`