Question

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

Answer 1

A recursive definition for the sequence {1, 2, 2, 4, 8, 32} starts off with initial terms and continues with a pattern using multiplication by increasing powers of 2.

Step-by-step explanation:

The question involves finding a recursive definition for the sequence {1, 2, 2, 4, 8, 32}. A recursive definition defines the terms of a sequence based on the previous terms. Observing the given sequence, it seems like the second term repeats and then every term from there on is the result of multiplying the previous term by an increasing even integer (2, 4, 8). Therefore, two different recursive formulas could be at play, one for the repeated term and one for the multiplicative pattern that follows.

Here is a potential recursive definition:

a(1) = 1 (the first term is 1)

a(2) = 2 (the second term is 2, starting the multiplicative pattern)

For n > 2, a(n) = a(n - 1) * 2^(n - 3) (this starts the pattern of multiplication from the third term onwards)

Note that the exponent starts at 0 for n = 3 (since 2^0 = 1) and increases by 1 for each subsequent term; hence, the '-3' in the formula to adjust the starting point of the exponent.

Answer 2

Given that the sequence is:
1,2,4,8,32
the recursive formula will be found as follows:
first term is=1
the sequence can be written as:
1,2,4,8,32
=(1*2^0),(1*2^1),(1*2^2),(1*2^3),(1*2^4)
thus the recursive formula will be
an=a1(r)^(n-1)
plugging the values we get:
an=1(2)^(n-1)