Find solutions in natural numbers

[anemone]

Find the solutions in natural numbers for the following equation:

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

 

[Greg]

Find the solutions in natural numbers for the following equation:

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

Spoiler

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 $x=1$.

 

[topsquark]

Spoiler

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

 

[anemone]

Good job, greg1313!(Cool)