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quick maths question

Solve the recurrence relations:
(i) un+1 = 3un + 4, n = 0, 1, 2, ...;

can someone expalin the steps thanks

I'm pretty sure that you'd be most likely to be given a formula like:

un+1 = pun + q

and then asked to find the values of p and q if the given values of n are 0, 1 and 2

Simultanous equations. You take your first two values of n and plug them into the equation. In this case you would plug in 0 and 1. This will give you your first formula. Then plug in the other two values, 2 and 1, this gives you the 2nd equations. Simltaneous equations, solve for p. Plug the value of p into one of the equations and solve for q.

thanks!!

actulyy youve kinda changed the quest round, does any one no the answer to the above question

Reply 4

gyrase

kevins

Solve the recurrence relations:
(i) un+1 = 3un + 4, n = 0, 1, 2, ...;

can someone expalin the steps thanks

u_1 = 3(0) + 4 = 4
u_2 = 3(1) + 4 = 7
u_3 = 3(2) + 4 = 10

4, 7, 10, an arithemetic progression with first term 4 and common difference +3.

Reply 5

kevins

would i then put the 4 and 3 in the ap formula? and would that then be my answer. thanks

Reply 6

Mush

but your numbers for n don't work. Because the second value for n is derived from plugging in the first value. ie.

1 = 3(0) + 4
1 = 4 ...which is obviously impossible:|

Reply 7

yusufu

Mush

but your numbers for n don't work. Because the second value for n is derived from plugging in the first value. ie.

1 = 3(0) + 4
1 = 4 ...which is obviously impossible:|

u₀, u₁, u₂ mean term 0, 1, 2 etc

Reply 8

Mush

ah....sorry my mistake! Well at least you know what to do should you be given the equation

Un+1 = pUn + q

and asked to find values for p and q :smile:

Reply 9

kevins

what would the actual answer be then?

Reply 10

yusufu

kevins

Solve the recurrence relations:
(i) un+1 = 3un + 4, n = 0, 1, 2, ...;

can someone expalin the steps thanks

If you mean work out an equation that tells you each term, we'll need to know u₀

Reply 11

kevins

lol, the question just says solve and no Uo is given :s-smilie:

any other ideass?

Since we don't know U(0), we'll introduce an arb constant in terms of U(0) to give a general solution (ie treat this as you'd treat a differential equation)

1) Look for a complementary function

since U(n+1) - 3U(n) = 4 we'll first solve U(n+1) - 3U(n) = 0

sub in U(n) = ar^n

then ar^(n+1) - 3ar^n = 0

giving r = 3

so CF is U(n) = a*3^n

2) Find a particular integral. Since all other terms are constants

try U(n) = p

then

p - 3p = 4

so p =-2

so our general solution is

U(n) = a*3^n - 2

for arbitrary a

what this means is that for whatever U(0) we choose, we get a = U(0) +2

then the formula U(n) = [U(0) + 2]*3^n - 2 will give all other terms. Try it!