A recursive sequence is defined by - d k = 6 d k − 1 + 3 , for all integers k ≥ 2 and d1 = 2 Use iteration to guess an
A recursive sequence is defined by -
d k = 6 d k − 1 + 3 , for all integers k ≥ 2
and d1 = 2
Use iteration to guess an explicit formula for the above sequence.
The above given question needed little correction as it is not satisfy the recursive function.
solution:
First, we change the subscripts by one because d starts with starts with and not
. Let Then
, for all integers k1
=2
Then we use iteration and try to guess the explicit formula or D:
we observe that the difference of these numbers is a multiple of 5. More specially the above numbers can be rewritten as follows:
we can easily see that the sequence of multiplicands 0,1,3,7,15 is described by the formula . so we claim that
for all integers
we need to prove that claim by induction
Step by step
Solved in 2 steps
