By Induction $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ But I didn't understand how I can do this, for this function or another function, i.e how to derive the right hand side function for the summation.
6333
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other?
37972
I have a Fibonacci problem ... basically I have to show that:  $$F(n+2) = 1 + \sum_{i=0}^{n} F(i)$$  Using strong induction, I must now show that:  $$F(n+2) = F(n+1) + F(n)$$ $$F(n+2) = (1 + \sum_{i=0}^{n-1} F(i)) + (1 + \sum_{i=0}^{n-2} F(i))$$ $$F(n+2) = 2 + \sum_{i=0}^{n-1} F(i) +  \sum_{i=0}^{n-2} F(i)$$    How do I add these two summations to show that they equal the first equation (at the top)?  Please provide the steps with explanation as to what you are doing. Thanks
39877
How do I derive this using reduction formula?  $$I_n=\int{\frac{x^n}{\sqrt{x^2+a^2}} dx}$$  I got $\sqrt{x^2+a^2}(x^n) - \frac{1}{n-1}(x^{n-1})\sqrt{x^2+a^2} + \frac{1}{n-1}(I_n) $ . I'm not too sure if this is correct. I would appreciate any form of help or correction to my answer. Thank you.
9
I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing?
112056
I need to get a closed form from  $$ \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1 $$  Starting from the most outer summation, I got  $$ \sum_{k=1}^{j} 1 = j $$  But now I don't know how to proceed with:  $$ \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} j $$  Could you guys please help me?  Thanks in advance.
30701
$$\sum_{i=1}^n \frac1{n+i} = \sum_{i=1}^{2n}\frac{(-1)^{i+1}}i,\quad \text{ for }n\ge 1.$$  I really have no clue how on to solve this one. It's the last one on my assignment. Would really enjoy some help with this one.
15333
How to prove that $ 2&lt;(1+\frac{1}{n})^{n}$ for every integer $n&gt;1$. I was thinking by induction, it works for $n=2$ but then I couldn't move forward.
85800
I'm having trouble proving this by induction. We need to show that $P(k+1)$ is true: $$\sum_{i=2}^{k+1} \frac{1}{i^2}\leq 1.$$  Don't know where to go from here. Any help?
73816
Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated.
85851
Prove that for every integer n>=2:  We have the summation of   $$ \frac{1}{i(i-1)}=1-\frac{1}{n} $$  I tried the algebra with this proof, but couldn't get it. I know that you split the i-1 and the i up but I am stuck from there.
36011
$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that  $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$  I can't seem to get this answer. Can someone please explain how to get to this?   Thanks in advance.
115435
I know that  $$\sum^n_{i=1}i^2=\frac{1}{6}n(n+1)(2n+1)$$ and  $$\sum^n_{i=1}i^3=\left(\sum^n_{i=1}i\right)^2.$$ Here is the question: is there a formula for  $$\sum^n_{i=1}i^4.$$
57506
I've seen the identity $\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ used here recently.  I checked for proofs here   I couldn't have figured out the combinatorial proof by myself, and the inductive proof assumes you already know the answer...  So my question is : do you know how to prove directly through computation that $\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ ?
3638
Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
90968
Must show:     $\sum_{i=0}^n i{n \choose i} = n2^{n-1}$ using a combinatorial argument. I have tried picking the problem apart by determining that the summation is equal to 1/2 of the size of all subsets of a set of size n times n.  But I am having trouble relating that to what each term of the summation represents.
67934
$$\frac{\sum_{i=0}^nx_iy_i-\bar y\sum_{i=0}^nx_i}{\sum_{i=0}^nx_i^2-\bar x\sum_{i=0}^nx_i}= \frac{\sum_{i=0}^n(x_i-\bar x)(y_i-\bar y)}{\sum_{i=0}^n(x_i-\bar x)^2}$$  Maybe it refers some calculation rules about summation. I cannot solve that. Thanks!
52877
So I've been struggling with this sum for some time and I just can't figure it out. I tried proving by induction that if the sum above is a $S_n$ then $S_{n+1} = 4S_n$, but I didn't really succeed so here I am. Thanks in advance.