MHB Proof by Induction - in Sequences.

AI Thread Summary
The discussion focuses on a sequence defined by the recurrence relation b_n = b_{n-1} + 2b_{n-2} with initial conditions b_1 = 1 and b_2 = 5. Participants calculate the first ten terms of the sequence, which are 1, 5, 7, 17, 31, 65, 127, 257, 511, and 1025. The main task involves proving the closed form b_n = 2^n + (-1)^n using strong induction, confirming it holds for n=1 and then for n+1 based on the assumption for n and n-1. An alternative method to derive the closed form through characteristic equations is also discussed, leading to the same result. The thread emphasizes the importance of both induction and alternative approaches in solving recursive sequences.
johnny009
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Dear ALL,

My last Question of the Day?

Let b1 and b2 be a sequence of numbers defined by:

$$b_{n}=b_{n-1}+2b_{n-2}$$ where $b_1=1,\,b_2=5$ and $n\ge3$

a) Write out the 1st 10 terms.

b) Using strong Induction, show that:

$$b_n=2^n+(-1)^n$$

Many Thanks

John C.
 
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Interesting question. I am puzzled as to why you have shown no attempt at all to do anything on this yourself. Don't you think it is interesting?

You are given the recursive definition: $b_n= b_{n-1}+ 2b_{n-2}$ with $b_1= 1$ and $b_2= 5$.

1. Write out the first ten terms.
$b_1= 1$
$b_2= 5$
$b_3= b_2+ 2b_1= 5+ 2(1)= 7$
$b_4= b_3+ 2b_2= 7+ 2(5)= 17$
$b_5= b_4+ 2b_3= 17+ 2(7)= 31$
$b_6= b_5+ 2b_4= 31+ 2(17)= 65$
$b_7= b_6+ 2b_5= 65+ 2(31)= 127$
$b_8= b_7+ 2b_6= 127+ 2(65)= 257$
$b_9= b_8+ 2b_7= 257+ 2(127)= 511$
$b_{10}= b_9+ 2b_8= 511+ 2(257)= 1025$

2. Using strong induction show that $b_n= 2^n+ (-1)^n$ for all $n\ge 1$.
I thought I recognized some of those numbers! 1= 2- 1, 5= 4+ 1, 7= 8- 1, 17= 16+ 1, 31= 32- 1, etc.

Strong induction requires proving that statement P(n)
1) is true for n= 1
2) if P(k) is true for all $k\le n$ then P(n+1) is true.

That is sufficient to conclude "P(n) is true for all $n\ge 1$".

Here, P(n) is "$b_n= 2^n+ (-1)^n$". P(1) is $b_1= 2^1- 1= 1$ which is true.

Now, suppose the statement is true for all $k\le n$. Then it is, specifically, true for n and n- 1.
We know that $b_{n+1}= b_{n}+ 2b_{n-1}= 2^n+ (-1)^n+ 2(2^{n-1}+ (-1)^{n-1})$.

So $b_{n+1}= 2^n+ (-1)^n+ 2^n+ (-1)^{n-1}= 2(2^n)+ (-1)^n+ 2(-1)^{n-1}$.

Look at two cases:
1) n is odd. Then n-1 is even so $(-1)^n= -1$ and $(-1)^{n-1}= 1$. $(-1)^n+ 2(-1)^{n-1}= -1+ 2= 1$.
2) n is even. Then n- 1 is odd so $(-1)^n= 1$ and $(-1)^{n-1}= -1$. $(-1)^n+ 2(-1)^{n-1}= 1- 2= -1$.
So if n is odd, $b_{n+1}= 2(2^n)+ 1= 2^{n+1}+ (-1)^{n+ 1}$ and if n is even $b_{n+1}= 2(2^n)- 1= 2^{n+1}+ (-1)^{n+1}$.

In either case, $b_{n+1}= 2^{n+1}+ (-1)^{n+1}$ and we are done.
 
johnny009 said:
Dear ALL,

My last Question of the Day?

Let b1 and b2 be a sequence of numbers defined by:

$$b_{n}=b_{n-1}+2b_{n-2}$$ where $b_1=1,\,b_2=5$ and $n\ge3$

a) Write out the 1st 10 terms.

b) Using strong Induction, show that:

$$b_n=2^n+(-1)^n$$

Many Thanks

John C.


Another way to find the closed form (I know induction is required for the assignment, but I post this just to show another method) would be to write the given recursion as:

$$b_{n}-b_{n-1}-2b_{n-2}=0$$

We have a linear homogeneous recursion, whose associated characteristic equation is:

$$r^2-r-2=(r-2)(r+1)=0$$

As the roots are:

$$r\in\{-1,2\}$$

We know the closed form will be:

$$b_{n}=c_12^n+c_2(-1)^n$$

Where the parameters $c_i$ can be determined from the given initial conditions:

$$b_{1}=c_12^1+c_2(-1)^1=2c_1-c_2=1$$

$$b_{2}=c_12^2+c_2(-1)^2=4c_1+c_2=5$$

Solving the 2X2 system, we determine:

$$\left(c_1,c_2\right)=(1,1)$$

Hence, the solution satisfying all given conditions is:

$$b_{n}=2^n+(-1)^n$$
 
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