Determine first five terms of sequence?

Determine the first five terms of the sequence. See image for the problem.
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a1=2 and a2=3. What are the other three numbers in the first five terms of the sequence?

1 Answer
George C.
Dec 7, 2017

#2, 3, 22, 103, 522#

Bonus: #a_n = -7/6(-1)^n+1/6(5)^n#

Explanation:

Given:

#{ (a_1 = 2), (a_2 = 3), (a_n = 4a_(n-1)+5a_(n-2)) :}#

The first #5# terms are:

#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# must satisfy:

#r^2-4r-5 = 0#

That is:

#(r+1)(r-5) = 0#

So #" "r = -1" "# or #" "r = 5#

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 #a_1# and #a_2#, we find:

#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 #30B=5# and hence #B=1/6#

Then using this value for #B# in the first equation, we find:

#A=5B-2 = 5/6-2 = -7/6#

So:

#a_n = -7/6(-1)^n+1/6(5)^n#

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