Simplifying Mathematical Induction for Sequences Using Algebra

[foreverdream]

I have an idea of what to do and I have reached the stage but when i am at final stage- I am struggling to simplify as i simply don't understand

(1/2)+(2/2^2) +(3/2^3)+...+ (n/2^n) = 2 - (n+2/2^n)

so I have done p(k) and p(k+1)

this gives me p(k)
(1/2)+(2/2^2) +(3/2^3)+...+ (k/2^k) = 2 - (k+2/2^k)

Next for p(k+1)
I get
(1/2)+(2/2^2) +(3/2^3)+...+ (k+1/2^k+1)

=2 - (k+2/2^k)+(k+1/2^k+1) by p(k)

now I am stuck in doing algebra.....?

 

[vela]

I have an idea of what to do and I have reached the stage but when i am at final stage- I am struggling to simplify as i simply don't understand

(1/2)+(2/2^2) +(3/2^3)+...+ (n/2^n) = 2 - (n+2/2^n)

so I have done p(k) and p(k+1)

this gives me p(k)
(1/2)+(2/2^2) +(3/2^3)+...+ (k/2^k) = 2 - (k+2/2^k)

Next for p(k+1)
I get
(1/2)+(2/2^2) +(3/2^3)+...+ (k+1/2^k+1)

=2 - (k+2/2^k)+(k+1/2^k+1) by p(k)

now I am stuck in doing algebra.....?

Yes, it's just algebra now. You want to show that from what you have, it follows that $$\frac{1}{2} + \frac{2}{2^2} + \cdots + \frac{k+1}{2^{k+1}} = 2 - \frac{(k+1)+2}{2^{k+1}}.$$

 

[foreverdream]

Yes I know but I can't seem to materialise step showing that final out come.

 

[vela]

Just combine the last two terms:
$$-\frac{k+2}{2^k}+\frac{k+1}{2^{k+1}} =\ ?$$

 

[foreverdream]

that simplification is what i am getting wrong

 

[vela]

Show your work.

 

[foreverdream]

Ok here it is:
2- (k+1)/2^k + (k+2)/2^k+1

=2-( 2^k+1(k+1)+(k+2)2^k )/ 2^k(2^k+1)
=2- 2^k+k+1 + 2^k+k + 4^k and then denominator as above. After this it doesn't make sense

 

[vela]

Ok here it is:
2- (k+1)/2^k + (k+2)/2^k+1

=2-( 2^k+1(k+1)+(k+2)2^k )/ 2^k(2^k+1)

You need to use parentheses to make clear what you mean. I think you meant
$$2 - \frac{k+1}{2^k} + \frac{k+2}{2^{k+1}} = 2 - \frac{2^{k+1}(k+1) + (k+2)2^k}{2^k 2^{k+1}} $$which is correct. I'm not sure what you did next, though.

=2- 2^k+k+1 + 2^k+k + 4^k and then denominator as above. After this it doesn't make sense

What was your thinking here? It looks like you tried to combine the factors out front with the exponents, which you can't do.

One of the rules of exponents is
$$x^a x^b = x^{a+b}$$If you have the product of some number x taken to two powers a and b, you can combine it into x taken to the sum a+b of those powers — or vice versa. Use that rule on the 2^k+1 in the numerator, and then pull out the common factor from the two terms in the numerator.

 

[foreverdream]

Don't worry I sorted it. Thanks for your help.