Think about powers of 7.
Hint:
The sequence {bn} is defined by b1 = 2, b2 = 8, b3 = 8bn-1 - 7bn-2 (n ≥ 3). Find b3 and b4 and guess a formula for bn. Use induction to prove your conjecture.
The first part of this is easy, b3 and b4 being 50 and 344 respectively, but what I am having trouble with is guessing a general formula and I was wondering if anybody could help me. I know how to prove it but I don't have anything to prove yet! Thanks a lot
Hello, slevvio!
The sequence is defined by: .
Find and and guess a formula for
Use induction to prove your conjecture.
There is an intricate algebraic procedure for determining the general term.
But as I started it, I eyeballed the first few terms ... and saw a pattern!
The first few terms are:
. .
Those values looked "kind of familiar" . . .
Then I recalled the powers-of-7 (which I had memorized years ago):
. .
Get it? . . . Those terms are one more than a power-of-7.
I think we've got it! . . .
I tested my conjecture with the next term: .
. . which just happens to be: . . . . Yay!