Binomial Theorem

Binomial Theorem: Level 3 Challenges


In the expansion of (2x+kx)8(2x+\frac{k}{x})^8, where kk is a positive constant, the term independent of xx is 700000700000. Find k.k.

r=0n(1)r(nr)1\large \displaystyle \sum _{ r=0 }^{ n } { (-1 )}^{ r } { \binom{n}{r} }^{-1}

If nn is an an odd positive integer, find the value of this sum.

(71+1)71(711)71 \left ( \sqrt {71} +1 \right )^{71} - \left ( \sqrt {71} -1 \right )^{71}

What is the last digit of the number above?

What is the integral part of number (2+1)6(\sqrt2+1)^6?

Details and Assumptions:

  • As an explicit example, the integral part of 123.456123.456 is 123123.

How many trailing zero(s) are there in the constant term of (x+1x)2014 \left (x+\frac{1}{x} \right )^{2014}?

Details and Assumptions:

  • The number of trailing zeroes in a number is the number of zeroes at the end of the number, e.g., 100100 has 22 trailing zeroes, and 100000001100000001 has none.

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