Proposed during Binomial era

Algebra Level 5

Evaluate

2018 2018 − ∑ r = 1 2018 ( 2019 ) ( r − 1 ) 2018 r − 1 r ( 2018 r ) = ? \large {2018}^{ 2018 } - \sum _{ r=1 }^{ 2018}{ \frac {( 2019) ( r - 1) { 2018}^{ r - 1 } }{r \dbinom {2018}r } } = ?


The answer is 1.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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2 solutions

This is not a complete solution :

( m + 1 ) ( r − 1 ) m r − 1 r ( m r ) = m r ( m r ) − m r − 1 ( m r − 1 ) \dfrac{(m+1)(r-1)m^{r-1}}{r\dbinom{m}{r}}= \dfrac{m^r}{\dbinom{m}{r}}-\dfrac{m^{r-1}}{\dbinom{m}{r-1}}

Now, it is simple telescoping :)

8 Helpful 0 Interesting 0 Brilliant 0 Confused

@Priyanshu Mishra can u please send me the link of this aits question paper advanced ?

A Former Brilliant Member - 3 years, 3 months ago

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@Mayank Singhal , reg.fiitjee.com

Priyanshu Mishra - 3 years, 3 months ago

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I logged in but I am getting mains question paper . Can u provide me the exact link to the advanced paper?

A Former Brilliant Member - 3 years, 3 months ago

Hey, Nice question, But if you have given find the value of the expression for m=k (some integer), Then the person have to solve the question, or else, Some can apply value putting and it destroys the beauty of such questions.

Thankyou.

Md Zuhair - 3 years, 3 months ago
Vitor Juiz
Feb 20, 2018

To any m, the answer it's the same. Therefore, if m=1, we get the expression equals to 1.

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