How to Solve Complicated Summations ?

As you all know the of sum of first \(n\) natural numbers is given by the formula \(\dfrac{n(n + 1)}{2}\), the sum of their squares is given by the formula \(\dfrac{n(n + 1)(2n + 1)}{6}\), the sum of their cubes is given by the formula \(\left ( \dfrac{n(n + 1)}{2} \right )^2\). Using these three formulas we used to solve questions based on summation easily. But what if the expression which has to be summed has higher orders like \(n^4, n^5, etc\). You can derive the sum of \(4^{th}\) powers with some difficulty but it will be impossible to do it for fifth powers and so on. So, in that situations we are not going to solve the series using formulae rather we go by technique. The main concept we should remember while doing by this technique is that we should express every term of the given series as difference of two factors so that all terms will get cancelled except the first one and last one. There are two types of methods based on the given series. We will discuss about them more clearly here.


\(\Large \color{green} \mathcal{TYPE : 1}\)

The main idea in this method is, splitting the \(n^{th}\) term as a difference of two terms. This method can be applied to a series in which each term is composed of \(r\) factors which are in AP, the first factor of the several terms being in AP. An example for such series is given below : \[(1 \times 2 \times 3 \times 4) + (2 \times 3 \times 4 \times 5) + ...\] WORKING RULE :

\(\text{Step - 1 :}\) Write down the \(n^{th}\) term

\(\text{Step - 2 :}\) Multiply both the sides with \(\boxed{{\color{blue}\text{Next Factor of last term}} - \color{pink}\text{Previous Factor of first term}}\) .

\(\text{Step - 3 :}\) Split the RHS and arrange them in such a way that all the diagonal elements gets cancelled. And there you have got the answer.

Here is an illustration to explain this more clearly.

Illustration - 1 : Find the sum of the given below series up to \(n\) terms. \[\large (1 \cdot 2 \cdot 3 \cdot 4) + (2 \cdot 3 \cdot 4 \cdot 5) + (3 \cdot 4 \cdot 5 \cdot 6) + \quad ...\]


Solution :

Our first job is find the \(n^{th}\) term. Here it is :

\(t_n = n(n + 1)(n + 2)(n + 3)\)

Now, we we should multiply both the sides with \(\text{Next Factor of last term - Previous Factor of first term} \implies \underbrace{(n + 4)}_{\text{Next factor of (n + 3)}} - \underbrace{(n - 1)}_{\text{Previous factor of n}}\)

\[[(n + 4) - (n -1)]t_n = n(n + 1)(n + 2)(n + 3)[(n + 4) - (n - 1)] \\ 5t_n = n(n + 1)(n + 2)(n + 3)(n + 4) - (n - 1)n(n + 1)(n + 2)(n + 3)\]

Now put \(n = 1\), we will get \(5t_1 = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 - 0\)

Now put \(n = 2\), we will get \(5t_2 = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 - 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5\)

Now put \(n = 3\), we will get \(5t_3 = 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 - 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6\)

And so on, now put \(n = (n - 1)\), we will get \(5t_{n - 1} = (n - 1) \cdot n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3) - (n - 2) \cdot (n - 1) \cdot n \cdot (n + 1) \cdot (n + 2)\)

And now put \(n = n\), we will get \(5t_n = n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3) \cdot (n + 4) - (n - 1) \cdot n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3)\)

Now, arrange each term in a separate line and see the magic !!! All the elements diagonally except first one and last will get cancelled while adding all the terms. \[\begin{array} ~\require{cancel} 5t_1 & = & {\color{blue}\cancel{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} & - & {\color{green}0} \\ 5t_2 & = & {\color{pink}\cancel{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}} & - & {\color{blue}\cancel{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}} \\ 5t_3 & = & {\color{red}\cancel{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7}} & - & {\color{pink}\cancel{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}} \\ 5t_4 & = & {\color{purple}\cancel{4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}} & - & {\color{red}\cancel{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7}} \\ & & & \vdots \\ 5t_{n - 1} & = & {\color{orange}\cancel{(n - 1) \cdot n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3)}} & - & {\color{teal}\cancel{(n - 2) \cdot (n - 1) \cdot n \cdot (n + 1) \cdot (n + 2)}} \\ 5t_n & = & {\color{green}n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3) \cdot (n + 4)} & - & {\color{orange}\cancel{(n - 1) \cdot n \cdot (n + 1) \cdot (n + 2) \cdot (n + 3)}} \\ \hline \\ 5({\color{blue}t_1 + t_2 + t_3 + ... + t_n}) & = & n(n + 1)(n + 2)(n + 3)(n + 4) \\ 5{\color{blue}S_n} & = & n(n + 1)(n + 2)(n + 3)(n + 4) \\ & & \boxed{\color{green} S_n = \dfrac{n(n + 1)(n + 2)(n + 3)(n + 4)}{5}} \\ \end{array}\]

You can see how every term with similar colors (except green) will get cancelled throughout leaving us only with two terms left.

This is how you have to solve these types of problems. Hope you have felt this quite good enough so let us move to the next set of problems !!!


\(\Large \color{green} \mathcal{TYPE : 2}\)

In this type of series too the main idea is to split every term into factors so that every term gets cancelled. The difference between the before one and this one is that the terms will be in reciprocal. This method can be applied to a series in which each term is composed of the reciprocal of the product of 'r factors in AP, the first factor of the several terms being in the same AP. Here is an example : \[\dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6} + ...\] WORKING RULE :

\(\text{Step - 1 :}\) Write down the \(n^{th}\) term

\(\text{Step - 2 :}\) Multiply both the sides with \(\boxed{{\color{blue}\text{Last Factor}} - \color{pink}\text{First Factor}}\) .

\(\text{Step - 3 :}\) Split the RHS and arrange them in such a way that all the diagonal elements gets cancelled. And there you have got the answer.

Here is an illustration to explain this more clearly.

Illustration - 2 : Find the sum of given below series up to n terms. \[\dfrac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \dfrac{1}{2 \cdot 3 \cdot 4 \cdot 5} + \dfrac{1}{3 \cdot 4 \cdot 5 \cdot 6} + ...\]


Solution :

Our first job is find the \(n^{th}\) term. Here it is :

\(t_n = \dfrac{1}{n(n + 1)(n + 2)(n + 3)}\)

Now, we we should multiply both the sides with \(\text{Last Factor - First Factor} \implies \underbrace{(n + 3)}_{\text{Last factor}} - \underbrace{n}_{\text{First factor}}\)

\([(n + 3) - n]t_n = \dfrac{(n + 3) - n}{n(n + 1)(n + 2)(n + 3)} \implies 3t_n = \dfrac{n + 3}{n(n + 1)(n + 2)(n + 3)} - \dfrac{n}{n(n + 1)(n + 2)(n + 3)}\)

\(\implies 3t_n = \dfrac{1}{n(n + 1)(n + 2)} - \dfrac{1}{(n + 1)(n + 2)(n + 3)}\)

Now put \(n = 1\), we will get \(3t_1 = \dfrac{1}{1 \cdot 2 \cdot 3} - \dfrac{1}{2 \cdot 3 \cdot 4}\)

Now put \(n = 2\), we will get \(3t_2 = \dfrac{1}{2 \cdot 3 \cdot 4} - \dfrac{1}{3 \cdot 4 \cdot 5}\)

Now put \(n = 3\), we will get \(3t_3 = \dfrac{1}{3 \cdot 4 \cdot 5} - \dfrac{1}{4 \cdot 5 \cdot 6}\)

And so on now put, \(n = (n - 1)\), we will get \(3t_{n - 1} = \dfrac{1}{(n - 1) \cdot n \cdot (n + 1)} - \dfrac{1}{n \cdot (n + 1) \cdot (n + 2)}\)

And now put, \(n = n\), we will get \(3t_n = \dfrac{1}{n \cdot (n + 1) \cdot (n + 2)} - \dfrac{1}{(n + 1) \cdot (n + 2) \cdot (n + 3)}\)

Now, arrange each term in a separate line and see the magic !!! All the elements diagonally except first one and last will get cancelled while adding all the terms. \[\begin{array} ~ \require{cancel} 3t_1 & = & {\color{green}\dfrac{1}{1 \cdot 2 \cdot 3}} & - & {\color{blue}\cancel{\dfrac{1}{2 \cdot 3 \cdot 4}}} \\ 3t_2 & = & {\color{blue}\cancel{\dfrac{1}{2 \cdot 3 \cdot 4}}} & - & {\color{pink}\cancel{\dfrac{1}{3 \cdot 4 \cdot 5}}} \\ 3t_3 & = & {\color{pink}\cancel{\dfrac{1}{3 \cdot 4 \cdot 5}}} & - & {\color{red}\cancel{\dfrac{1}{4 \cdot 5 \cdot 6}}} \\ 3t_4 & = & {\color{red}\cancel{\dfrac{1}{4 \cdot 5 \cdot 6}}} & - & {\color{purple}\cancel{\dfrac{1}{5 \cdot 6 \cdot 7}}} \\ &&& \vdots \\ 3t_{n - 1} & = & {\color{teal}\cancel{\dfrac{1}{(n - 1) \cdot n \cdot (n + 1)}}} & - & {\color{orange}\cancel{\dfrac{1}{n \cdot (n + 1) \cdot (n + 2)}}} \\ 3t_n & = & {\color{orange}\cancel{\dfrac{1}{n \cdot (n + 1) \cdot (n + 2)}}} & - & {\color{green}\dfrac{1}{(n + 1) \cdot (n + 2) \cdot (n + 3)}} \\ \hline \\ 3({\color{blue}t_1 + t_2 + t_3 + ... + t_n}) & = & \dfrac{1}{6} & - & \dfrac{1}{(n + 1)(n + 2)(n + 3)} \\ 3S_n & = & \dfrac{1}{6} & - & \dfrac{1}{(n + 1)(n + 2)(n + 3)} \\ & & \boxed{\color{green} S_n = \dfrac{1}{18} - \dfrac{1}{3(n + 1)(n + 2)(n + 3)}} \\ \end{array}\]

You can see how every term with similar colors (except green) will get cancelled throughout leaving us only with two terms left. In general I solved the problems taking the number of terms as \(n\) but you can find the sum by taking number of terms as \(n\) and at the ant substitute the value of \(n\) as according to the question.


This is how we have to solve these types of series so now it's your turn to practice them. If you have any doubts you can keep them in the comments section below.

Want to improve in Mathematics !!! Want to see more of these !!! Then try my set Mathematics Done Right

\[\Large \color{teal} \mathcal{Thank \ You \ !!!}\]

Note by Ram Mohith
1 month ago

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@Ram Mohith, For the sum of cubes I think that

\[1^3 + 2^3 + 3^3 \cdots n^3 = (1 + 2 + 3 \cdots n)^2\]

You do not really have to memorise the formula

Mohmmad Farhan - 4 weeks, 1 day ago

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Yes you are correct. \[\displaystyle \sum_{x = 1} ^n x^3 = \left ( \displaystyle \sum_{x = 1} ^n x \right )^2\]

Ram Mohith - 4 weeks, 1 day ago

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I like this note!

Mohmmad Farhan - 1 month ago

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Thank You

Ram Mohith - 1 month ago

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I made an inspirational note about the Collatz Conjecture

Mohmmad Farhan - 1 month ago

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You know I really worked hard on writing such a complicated latex code !!!

Ram Mohith - 1 month ago

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I have experienced this. See my note:\(\pi\), a beautiful number

Mohmmad Farhan - 1 month ago

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@Mohmmad Farhan Yes I have seen it. It is a good one

Ram Mohith - 1 month ago

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@Ram Mohith How was the vacation? Did it go well?

Mohmmad Farhan - 1 month ago

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@Mohmmad Farhan Yes good. I planned to post this note before my vacation but it was not completed by then. So, today I finished it off.

Ram Mohith - 1 month ago

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@Ram Mohith Did you join, Jason Dyers group? It is very useful (if you use it well)

Mohmmad Farhan - 1 month ago

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@Mohmmad Farhan Open problem group. I have joined it already.

Ram Mohith - 1 month ago

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@Ram Mohith Ok!

Mohmmad Farhan - 1 month ago

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@Ram Mohith My thanks to you.

Mohmmad Farhan - 1 month ago

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@Ram Mohith,

I FINALLY GOT A CONTRIBUTIONS PAGE! I AM SO EXCITED!

Dear Farhat.

Mohmmad Farhan - 2 weeks, 6 days ago

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Congratulations. And keep it up. Now itself you have got 496 comment upvotes which is 8 upvotes less than mine. Looks like you will cross my contributions page soon.

Ram Mohith - 2 weeks, 6 days ago

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What is a contributions page?

Ekene Franklin - 2 weeks, 6 days ago

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Awesomely wrote with Latex. Keep it up

Md Zuhair - 3 weeks, 2 days ago

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@Calvin Lin, where is the save option?

Mohmmad Farhan - 4 weeks, 1 day ago

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Can you send me a screenshot of what you see? The save function is still there.
I've tested this on other accounts / browsers, and have been unable to reproduce your issue.

Calvin Lin Staff - 3 weeks, 6 days ago

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@Calvin Lin, when it is on display mode(Thread), there is no save function but when I refresh the page there is a save function

Mohmmad Farhan - 3 weeks, 6 days ago

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Very great note, brother. Very informative to those ignorant of this.

Ekene Franklin - 4 weeks, 1 day ago

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Yes. Thank you very much.

Ram Mohith - 4 weeks, 1 day ago

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Please have a look at Faulhaber polynomials. Finding general expressions for sums of \(k^{\text{th}}\) powers is not particularly difficult if you are familiar with certain properties of it, as well as the property of Bernoulli numbers.

Gennady Notowidigdo - 4 weeks, 1 day ago

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I agree with Ram Mohith. Not everyone has a rapport with Faulhaber's polynomials. Besides, Ram has done a great work on the LaTeX.

Ekene Franklin - 4 weeks, 1 day ago

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Please see my reply somewhere in this thread. Given anyone can click a few buttons, I think repeating what I have said here would be a pointless exercise.

Gennady Notowidigdo - 4 weeks, 1 day ago

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I second you!

Mohmmad Farhan - 4 weeks, 1 day ago

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@Mohmmad Farhan Yes, Farhat. For example, I am twelve, and have a weak ability in Faulhaber's formula's. Besides, Nigerians don't have a unique passion for math.

Ekene Franklin - 4 weeks, 1 day ago

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@Ekene Franklin Math feeds us but sometimes it is out of our grasp

Mohmmad Farhan - 4 weeks, 1 day ago

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@Mohmmad Farhan That is a true saying, Farhat, and is worthy of accolade.

Ekene Franklin - 4 weeks, 1 day ago

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@Ekene Franklin Math knowledge is indeed fueled by passion

Mohmmad Farhan - 4 weeks, 1 day ago

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@Mohmmad Farhan Very correct, Farhat. Intelligent oration there.

Ekene Franklin - 4 weeks, 1 day ago

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@Ekene Franklin My thanks to you

Mohmmad Farhan - 4 weeks, 1 day ago

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@Mohmmad Farhan No problem, friend.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin MATH - Means, Always, The Heaven

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan True but not more than science for me.

Ram Mohith - 4 weeks ago

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@Ram Mohith Oh yeah! Science is the curiosity and wonders that is the size of whole worlds.

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan Math and science are the heartbeat of the universe. They oxidise our world to a better place, reducing ignorance in the process. They catalyse the increase of knowledge. They adorn our lives to make them frictionless. Oh, what is better than math and science !!!

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Exceptionally excellent. Should be a Master peice.

Ram Mohith - 4 weeks ago

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@Ram Mohith Thank you, brother.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Humanity is full of brothers and sisters. Not only connected by blood but also heart

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan Two young dynamites.

Ram Mohith - 4 weeks ago

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@Ram Mohith @Ram Mohith, Did you know that dynamites mean bombs (a type of bomb) that were discovered by Alfred Nobel (Nobel Prize creator)

Mohmmad Farhan - 4 weeks ago

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@Ekene Franklin WOW! You have great poetry skills

Mohmmad Farhan - 4 weeks ago

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@Ekene Franklin I think you make a great poet for my math poetry contest

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan Ok. But I don't understand the rules.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Which part of the rules?

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan The truth is that I don't know how to create a link.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Oh! The structure is like [link text ]then you add (link) (\([link text](link)\))

Note: I added the latex code to prevent it from rendering but do not add the latex code

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan Ok. Something like EKENE FRANKLIN?

Ekene Franklin - 4 weeks ago

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@Ekene Franklin I think some mistake happened. When I clicked it a page opens saying page not found. Check whether the link is correct or not. I prefer you to copy and paste the link rather than typing it so that there will be no errors.

Ram Mohith - 4 weeks ago

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@Ram Mohith Let me try again. @Ram Mohith.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin It is not still working.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin The link is the URL you see above this website

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan You mean brilliant.org?

Ekene Franklin - 4 weeks ago

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@Ekene Franklin So @EKENE FRANKLIN!

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Tap on that link

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan It is taking me to the home page!

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Because you only included Brilliant.org. you have to copy the URL wholesale! (Copy - Paste would help)

Mohmmad Farhan - 4 weeks ago

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@Mohmmad Farhan Ok.

Ekene Franklin - 4 weeks ago

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@Ekene Franklin Now, do you understand?

Mohmmad Farhan - 4 weeks ago

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@Ekene Franklin Yes but you have to be very specific. (Brilliant.org is only part of the URL)

Mohmmad Farhan - 4 weeks ago

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I have posted this for those who deosn't know those things.

Ram Mohith - 4 weeks, 1 day ago

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Still, nothing wrong with alternate perspectives... you should always be open to other ideas, even from the most advanced of sources.

There is a relation between these sums and the inverse of a triangular matrix related to Pascal's triangle. I find this method far more accessible to people, as opposed to your approach, since the required tools are easily known to enthusiastic students; moreover, it captures the main essence of these sums, as studied by Faulhaber and Bernoulli himself. Furthermore, Conway has done a lot of work in this area, where he is able to relate the Derivative/Integral linear mapping of these sums with lower/higher-degree ones of the same form; this is quite a recent result from a paper of his in 2016, which I assume has been published (I do not recall the source directly, unfortunately).

Gennady Notowidigdo - 4 weeks, 1 day ago

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People who are like me! (I am 10)

Mohmmad Farhan - 4 weeks, 1 day ago

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