SOLUTION: yn= 1.05y(n-1) +2 and y(0) = 3. Find an explicit formula for y(n).

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Question 1188459: yn= 1.05y(n-1) +2 and y(0) = 3. Find an explicit formula for y(n).
Found 3 solutions by MathLover1, mccravyedwin, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

y%28n%29=+1.05y%28n-1%29+%2B2 and y%280%29+=+3.
Find an explicit formula for y%28n%29.
y%28n%29=+1.05y%28n-1%29+%2B2..........if y%280%29+=+3
3=+1.05y%280-1%29+%2B2
%283-2%29%2F+1.05=y%28-1%29+
%283-2%29%2F+1.05=-y
20%2F21=-y
y=-20%2F21

y%28n%29=+1.05%28-20%2F21%29%28n-1%29+%2B2...........since 1.05%28-20%2F21%29=-1

y%28n%29=+-%28n-1%29+%2B2




Answer by mccravyedwin(408) About Me  (Show Source):
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
The other solution is incorrect.

A well-known theorem states that the general (explicit) formula for the recursion formula 

y%5Bn%5D%22%22=%22%22A%2Ay%5Bn-1%5D%2BB has the form y%5Bn%5D%22%22=%22%22C%2AA%5En+%2B+D

We calculate y(1) = 5.15, y(2) = 7.4075, substitute it and A=1.05

5.15%22%22=%22%22C%2A1.05%5E1+%2B+D

D%22%22=%22%225.15-1.05C

We calculate y(2) = 7.4075, substitute it and A=1.05

7.4075%22%22=%22%22C%2A1.05%5E2+%2B+D

7.4075%22%22=%22%221.1025C+%2B+D

D%22%22=%22%227.4075-1.1025C

Equating the expressions for D

5.15-1.05C%22%22=%22%227.4075-1.1025C

0.0525C%22%22=%22%222.2575

C%22%22=%22%2243

D%22%22=%22%225.15-1.05%2843%29

D-40

General (explicit) formula

y%5Bn%5D%22%22=%22%2243%2A1.05%5EN-40

Edwin