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:

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."

answer: x=3 and y=4

step-by-step explanation: