Question

Which recursive definition could be used to generate the sequence {1, 2, 2, 4, 8, 32,...}?

A) a1 = 1, and an = 2(an-1) + 2
B) a1 = 1, and an = 2(an-1) + 2n
C) a1 = 1, a2 = 2, and an = (an-2)(an-1)
D) a1 = 1, a2 = 2, and an = 2(an-2)(an-1)

Answer 1

n being the rank and a(n) being the value of a at rank n


We notice that each term is the product of the 2 consecutive terms before it:

4 = 2x2

8 = 2x4

32 = 4x8


so the formula is a(n) = a(n-2) x a(n-1)

Answer 2

`a_1 = 1, a_2 = 2, \ and \ a_n= (a_(n-2))(a_(n-1))`

Explanation:

the sequence {1, 2, 2, 4, 8, 32,...}

We need to find the recursive formula

First term is 1, second term is 2 , third term is 2 and so on

From the sequence we can see that

first term times second term is 1 times 2 is 2

`1 \cdot 2 = 2`

`2 \cdot 2 = 4`

`2 \cdot 4 = 8`

`4 \cdot 8 = 32`

Recursive formula = multiply the previous 2 terms

`a_1 = 1, a_2 = 2, \ and \ a_n= (a_(n-2))(a_(n-1))`