Find the closed form solutions of the following recurrence relations with given initial conditions. use forward substitution or backward substitution as described in example 10 in the text. (a) an = -an-1, a0 = 5 (b) an = an-1 + 3, a0 = 1 (c) an = an-1 - n, a0 = 4 (d) an = 2nan-1, a0 = 3

a)

b)

c)

d)

Step-by-step explanation:

We solve this using backward or forward substitution.

a) We have this:

then:

for we have:

from this, we can see that is a solution for this recurrence relation, where . This is:

b) We have with . Then:

but at the same time

we have:

or

by the next:

We can see that the recurrence rule is:

this is

c)Note that:

taking all this we have to:

then:

this is:

d)We take . Then:

replacing we have:

Step-by-step explanation:

Me eather