MHB Find solutions in natural numbers

AI Thread Summary
The discussion centers on finding natural number solutions for the equation involving a series of fractions that sum to 5. Participants express enthusiasm for the problem, with one user congratulating another for their contributions. The equation features factorial terms in the numerators and products of linear expressions in the denominators. The focus remains on solving the equation accurately within the constraints of natural numbers. The thread highlights the collaborative effort in tackling complex mathematical problems.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Find the solutions in natural numbers for the following equation:

$$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\cdots+\frac{10\cdot 9\cdot 8 \cdots\cdot 3 \cdot 2 \cdot 1}{(x+10)(x+9)(x+8)\cdots(x+3)(x+2)(x+1)}=5$$
 
Mathematics news on Phys.org
anemone said:
Find the solutions in natural numbers for the following equation:

$$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\cdots+\frac{10\cdot 9\cdot 8 \cdots\cdot 3 \cdot 2 \cdot 1}{(x+10)(x+9)(x+8)\cdots(x+3)(x+2)(x+1)}=5$$

Let $S$ be the sum Let $x=1$. Then we have
$$S=\dfrac{1}{11}\sum_{n=1}^{10}n=5$$ so $x=1$ is a solution. Increasing $x$ will result in a smaller sum (as the denominators of the fractions will be larger), so the only solution is $x=1$.
 
greg1313 said:
Let $S$ be the sum Let $x=1$. Then we have
$$S=\dfrac{1}{11}\sum_{n=1}^{10}n=5$$ so $x=1$ is a solution. Increasing $x$ will result in a smaller sum (as the denominators of the fractions will be larger), so the only solution is $x=1$.
Awesome! (Bow)

-Dan
 
Good job, greg1313!(Cool)
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...

Similar threads

Replies
3
Views
989
Replies
14
Views
2K
Replies
24
Views
3K
Replies
7
Views
2K
Replies
2
Views
2K
Replies
5
Views
2K
Back
Top