A Weird Sum of Fractions!

Algebra Level 4

S = ∑ i = 1 n 1 ( a i ) ( a i + a i + 1 ) ( a i + a i + 1 + a i + 2 ) ⋯ ( a i + a i + 1 + … + a i + n − 2 ) \large{ S = \sum_{i=1}^n \dfrac{1}{(a_i)(a_i + a_{i+1})(a_i + a_{i+1} + a_{i+2}) \dotsm (a_i + a_{i+1} + \ldots + a_{i+n-2}) }}

If a 1 , a 2 , … , a n a_1, a_2, \ldots, a_n are real numbers with ∑ i = 1 n a i = 0 \sum_{i=1}^n a_i = 0 , and where a n + 1 = a 1 , a n + 2 = a 2 a_{n+1} = a_1, a_{n+2} = a_2 and so on.. assuming that the denominators are non-zero. Then find the value of S S upto three correct places of decimal.


The answer is 0.000.

Satyajit Mohanty by Satyajit Mohanty , 24, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Arun Avirachan
Aug 16, 2015

ai =0 .. multyply with this is also zero... so denominator become zero.. then s = 0. 00 ( limit tendsa to infinity)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I couldn't understand anything. Please post a proper solution.

Satyajit Mohanty - 5 years, 10 months ago

I cannot post a solution for some reason, so I am writing here only:

The main idea is to use induction on n. Assume that the problem is true for n = k, let a 1 = b 1 , a 2 = b 2 , . . . , a k − 1 = b k − 1 a_1 = b_1, a_2 = b_2, ..., a_{k - 1} = b_{k - 1} and a k = b k + b k + 1 a_k = b_k + b_{k + 1} .

Vedant Saini - 4 months, 1 week ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...