Determine first five terms of sequence?
Determine the first five terms of the sequence. See image for the problem.
a1=2 and a2=3. What are the other three numbers in the first five terms of the sequence?
Determine the first five terms of the sequence. See image for the problem.
a1=2 and a2=3. What are the other three numbers in the first five terms of the sequence?
1 Answer
Bonus:
Explanation:
Given:
{ (a_1 = 2), (a_2 = 3), (a_n = 4a_(n-1)+5a_(n-2)) :}
The first
a_1 = 2
a_2 = 3
a_3 = 4a_2+5a_1 = 4(color(blue)(3))+5(color(blue)(2)) = 12+10 = 22
a_4 = 4a_3+5a_2 = 4(color(blue)(22))+5(color(blue)(3)) = 88+15 = 103
a_5 = 4a_4+5a_3 = 4(color(blue)(103))+5(color(blue)(22)) = 412+110 = 522
Bonus - What is a formula for a general term of this sequence?
Focusing on the recurrence rule step:
a_n = 4a_(n-1)+5a_(n-2)
is there a geometric sequence which obeys this rule?
If so, then its common ratio
r^2-4r-5 = 0
That is:
(r+1)(r-5) = 0
So
Hence the general term of the given sequence must be expressible in the form:
a_n = A(-1)^n + B(5)^n
Using our values for
2 = a_1 = A(-1)^1+B(5)^1 = -A+5B
3 = a_2 = A(-1)^2+B(5)^2 = A+25B
Adding these two equations, we find
Then using this value for
A=5B-2 = 5/6-2 = -7/6
So:
a_n = -7/6(-1)^n+1/6(5)^n