Intereting Posts

What Mathematics questions can be better solved with concepts from Physics?
How to prove DeMorgan's law?
The security guard problem
Hopkins-Levitzki: an uncanny asymmetry?
A question about differentiation
Can the inverse of a function be the same as the original function?
Find all primes $p$ such that $\dfrac{(2^{p-1}-1)}{p}$ is a perfect square
A maximization problem
Adjoint of the covariant derivative on a Riemannian manifold
Chebyshev polynomial question
Simple doubt about complex numbers
Subspace Topology of a subset
The Ring Game on $K$
Finitely generated modules over artinian rings have finite length
Peano/Presburger axioms – “find” numbers lower or equal than another number

Considering the following formulae:

(i) $1+2+3+..+n = n(n+1)/2$

(ii) $1\cdot2+2\cdot3+3\cdot4+…+n(n+1) = n(n+1)(n+2)/3$

- Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
- Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
- Proving $\sum\limits_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$
- Eigenvectors of harmonic series matrix
- If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?
- Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.

(iii) $1\cdot2\cdot3+2\cdot3\cdot4+…+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4$

Find and prove a ‘closed formula’ for the sum

$1\cdot2\cdot3\cdot…\cdot k + 2\cdot3\cdot4\cdot…\cdot(k+1) + … + n(n+1)(n+2)\cdot…\cdot (k+n-1)$

generalizing the formulae above.

I have attempted to ‘put’ the first 3 formulae together but I am getting nowhere and wondered where to even start to finding a closed formula.

- Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$
- Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$
- Summation Symbol: Changing the Order
- Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $
- Algebra-sum of entries in each column of a sqaure matrix = constant
- Reference for a tangent squared sum identity
- ${{p^k}\choose{j}}\equiv 0\pmod{p}$ for $0 < j < p^k$
- Sum of combinations of the n by consecutive k
- Smoothstep sigmoid-like function: Can anyone prove this relation?
- Finding sum $\sum_{i=1}^n \frac1{4i^2-1}$

The pattern looks pretty clear: you have

$$\begin{align*}

&\sum_{i=1}^ni=\frac12n(n+1)\\

&\sum_{i=1}^ni(i+1)=\frac13n(n+1)(n+2)\\

&\sum_{i=1}^ni(i+1)(i+2)=\frac14n(n+1)(n+2)(n+3)\;,

\end{align*}\tag{1}$$

where the righthand sides are closed formulas for the lefthand sides. Now you want

$$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\;;$$

what’s the obvious extension of the pattern of $(1)$? Once you write it down, the proof will be by induction on $n$.

**Added:** The general result, of which the three in $(1)$ are special cases, is $$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\frac1{k+1}n(n+1)(n+2)\dots(n+k)\;.\tag{2}$$ For $n=1$ this is $$k!=\frac1{k+1}(k+1)!\;,$$ which is certainly true. Now suppose that $(2)$ holds. Then

$$\begin{align*}\sum_{i=1}^{n+1}i(i+1)&(i+2)\dots(i+k-1)\\

&\overset{(1)}=(n+1)(n+2)\dots(n+k)+\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\\

&\overset{(2)}=(n+1)(n+2)\dots(n+k)+\frac1{k+1}n(n+1)(n+2)\dots(n+k)\\

&\overset{(3)}=\left(1+\frac{n}{k+1}\right)(n+1)(n+2)\dots(n+k)\\

&=\frac{n+k+1}{k+1}(n+1)(n+2)\dots(n+k)\\

&=\frac1{k+1}(n+1)(n+2)\dots(n+k)(n+k+1)\;,

\end{align*}$$

exactly what we wanted, giving us the induction step. Here $(1)$ is just separating the last term of the summation from the first $n$, $(2)$ is applying the induction hypothesis, $(3)$ is pulling out the common factor of $(n+1)(n+2)\dots(n+k)$, and the rest is just algebra.

If you divide both sides by $k!$ you will get binomial coefficients and you are in fact trying to prove

$$\binom kk + \binom{k+1}k + \dots + \binom{k+n-1}k = \binom{k+n}{k+1}.$$

This is precisely the identity from this question.

The same argument for $k=3$ was used here.

Or you can look at your problem the other way round: If you prove this result about finite sums

$$\sum_{j=1}^n j(j+1)\dots(j+k-1)= \frac{n(n+1)\dots{n+k-1}}{k+1},$$

you also get a proof of the identity about binomial coefficients.

For a fixed non-negative $k$, let $$f(i)=\frac{1}{k+1}i(i+1)\ldots(i+k).$$

Then $$f(i)-f(i-1)=i(i+1)\ldots(i+k-1).$$

By telescoping,

$$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\sum_{i=1}^n\left(f(i)-f(i-1)\right)=f(n)-f(0)=f(n)$$

and we are done.

I asked exactly this question a couple of days ago, here:

Telescoping series of form $\sum (n+1)\cdot…\cdot(n+k)$

My favourite solution path so far is

to start with the hockey stick identity.

From (i), (ii) and (iii) it is reasonable to guess that your sum will be

$$n(n+1)\cdot…\cdot(n+k)/(k+1)$$

Try to prove this by induction.

- How to solve this to find the Null Space
- Smallest number with specific number of divisors
- Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$
- Proving $A \cap C = B \cap C$, but $ A \neq B$
- concepts which is present in metric space but not in topological space
- Why is the second derivative of an inflection point zero?
- Prove that a matrix is invertible
- nth convolved Fibonacci numbers of order 6 modulo m
- where did determinant come from?
- Is there a category theory notion of the image of an axiom or predicate under a functor?
- Given a matrix with non-negative real entries, can you algebraically prove that it has a non-negative eigenvalue?
- Show that $y_1$ and $y_2$ are not Linearly Independent
- Bounded sequence which is not convergent, but differences of consecutive terms converge to zero
- Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$
- $X$ Poisson distribution, $Y$ geometric distribution – how to find $P(Y>X)$?