Algebra

# Binomial Theorem: Level 3 Challenges

In the expansion of $(2x+\frac{k}{x})^8$, where $k$ is a positive constant, the term independent of $x$ is $700000$. Find $k.$

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

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

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

What is the last digit of the number above?

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

Details and Assumptions:

• As an explicit example, the integral part of $123.456$ is $123$.

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

Details and Assumptions:

• The number of trailing zeroes in a number is the number of zeroes at the end of the number, e.g., $100$ has $2$ trailing zeroes, and $100000001$ has none.
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