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

Answer:

(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)$n^{th} = n!$

$f(1) = 1$

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

Step-by-step 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 $n^{th}$

$n^{th} = n!$

$f(1) = 1$

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