3 math questions (summation, mathematical induction)

[misogynisticfeminist]

2 math questions (summation, mathematical induction)

I have 2 questions regarding summation and mathematical induction

2. Prove by mathematical induction
$$\sum^n_{r=1} \frac {1}{r(r+2)} = \frac {3}{4} - \frac {(2n+3)}{2(n+1)(n+2)}$$
i am now trying to prove that

$$3/4 - \frac{2(n+1)+3}{2(n+2)(n+3)} - \frac {1}{n+1}{n+3}$$

I seem to get caught up with the algebra and what i get is

$$\frac {2n^2+5n+1}{2(n+1)(n+2)(n+3)}$$. instead of $$\frac {2n^2+5n+3}{2(n+1)(n+2)(n+3)}$$

3. Prove by induction that $$\sum^n_{r=1}rx^{r-1}=\frac{1-(n+1)x^n+nx^{n+1}}{(1-x)^2}$$

again, I'm caught up with the algebra.

 

[whkoh]

I use a different method:

1) proof summation is true when n=2
2) consider n=k+1.
$$\sum^{k+1}_{r=1} \frac{1}{r(r+2)} = \frac{3}{4} - \frac{2(k+1)+3}{2(k+2)(k+3)}$$
$$= \sum^k_{r=1} \frac{1}{r(r+2)} + \mbox{(k+1)^{th}} \mbox{term}$$

Well, it works. Sorry if I'm not very helpful

 

[HallsofIvy]

"i am now trying to prove that
$$3/4 - \frac{2(n+1)+3}{2(n+2)(n+3)} - \frac {1}{n+1}{n+3}$$"

This makes no sense- it is neither an equation nor a statement- it is a formula. What are you trying to prove about it?

What you should be trying to prove is that
$$\frac {3}{4} - \frac {(2n+3)}{2(n+1)(n+2)}+ \frac{1}{(n+1)(n+3)} = \frac{3}{4}- \frac{(2n+5)}{2(n+2)(n+3)}$$
The left side of that is just the formula for the sum to n plus the n+1 term and the right side is the same formula for the sum to n+1.
Now to show they are the same: obviously the "3/4" cancel. Get the common denominator for the two fractions left on the left side:
$$-\frac {(2n+3)(n+3)}{2(n+1)(n+2)(n+3)}+ \frac{2(n+2)}{2(n+1)(n+2)(n+3)}$$
$$-\frac{(2n^2+ 9n+ 9)+(2n+4)}{2(n+1)(n+2)(n+3)}$$
$$-\frac{2n^2+7n+5}{2(n+1)(n+2)(n+3)}$$
In order to have the same denominator as on the right, we obviously need to cancel the "n+1" in the denominator and, sure enough, 2n²+ 7n+ 5= (n+1)(2n+ 5)!

As far as 2 is concerned, are you required to use induction? I notice that the left side is just the derivative of 1+ x+ x²+...+ xⁿ and I know that xⁿ⁺¹- 1= (x- 1)(1+ x+ x²+...+ xⁿ) so that
$$1+ x+ x^2+...+ x^n= \frac{x^n-1}{x-1}$$. You can get the result by using the quotient rule to differentiate the right side.

Of course, if you MUST us induction, then you need to show that
$$\frac{1-(n+1)x^n+nx^{n+1}}{(1-x)^2}+ (n+1)x^n= \frac{1-(n+2)x^2+(n+1)x^{n+2}}{(1-x)^2}$$

Again, concetrate on the right hand side. You will need to multiply that "(n+1)xⁿ" term by (1-x)²/(1-x)² and then combine the numerators.