# 2017 times a is a natural number

1. Dec 16, 2016

### giokrutoi

1. The problem statement, all variables and given/known data

find 2017 times A
2. Relevant equations

3. The attempt at a solution
the sum of last members denominator.is 2017 multiplied by 1008
the member before it is 2015 multiplied by 1008
before it 2015 times 1007
2013 times 1007
2013 times 2016
how can i move futher

2. Dec 16, 2016

### Staff: Mentor

Have you tried to write this equation in summation notation? The denominators can be written in a closed form, which the inverse can be taken of.

3. Dec 16, 2016

### Buffu

$$A = {1 \over 1+ 2} + {1 \over 1+ 2 +3 } +\cdots + {1 \over 1+ 2+ \cdots +2016}$$
$$\color {red}{A + 1 = {1\over 1} + {1 \over 1+ 2} + {1 \over 1+ 2 +3 } +\cdots + {1 \over 1+ 2+ \cdots +2016}+ \cdots }$$

Find the nth term of 'red thing', then do partial fraction decomposition. You will a series that is very easy to compute.

4. Dec 16, 2016

### haruspex

A good first step is to try to spot the pattern. With just one term, it is 1/3. What is the sum of the first two? Of the first three?...

5. Dec 16, 2016

Staff Emeritus
Haruspex has a good hint. A related hint is "about how big do you expect your answer to be?"

6. Dec 17, 2016

### Delta²

Just one reminder, I guess you have been taught that $1+2+...+k=\frac{k(k+1)}{2}$, because it should be easy to recognize that, but you don't even mention it in your post or at relevant equations.