Binomial Coefficients

Number Theory Level 4

For a positive integer n, let C n = 1 n + 1 ( 2 n n ) C_n=\dfrac{1}{n+1} \binom{2n}{n} , and S n = C 1 + C 2 + … + C n S_n=C_1 + C_2 + \ldots + C_n .

Let's say S n ≡ 1 ( m o d 3 ) S_{n} \equiv 1 \pmod{3} if and only if there exists a "certain digit" in the base 3 expansion of n + 1 n + 1 .

Find that "certain digit".


The answer is 2.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Pranjal Jain
Mar 18, 2015

S 1 ≡ 1 m o d 3 S_{1}\equiv 1\mod 3

( 1 + 1 ) = 2 (1+1)=2 in base 3 3 is 2 2

Thus, 2 is required digit

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Well , there's an even better method to it .

Since it's base 3 , so permissible digits are 0,1 and 2 and you've got three tries !

Lol , I was just kidding :)

How did you fare in today's(well yesterday ) exam ?

A Former Brilliant Member - 6 years, 2 months ago

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