Verify that $\sum^{8564}_{i=82} \binom{8564}{i} < 2^{8564}$

Question

I have to tell if the following inequality is true: $$\sum^{8564}_{i=82} \binom{8564}{i} < 2^{8564}$$

but how do I tackle that? I reckon the standard formula for calculate the value of the binomial coefficient is kind of useless in this case, the summation startles me. I feel really stuck, can anyone please throw any hint at me?

Answer 1

Observe that $$2^n=(1+1)^n=\sum_{k=0}^n\binom{n}{k}1^k1^{n-k}=\sum_{k=0}^n\binom{n}{k}$$

Answer 2

Hint $$2^n = (1+1)^n = \sum_{i=0}^n \begin{pmatrix} n \\ i\end{pmatrix} 1^i 1^{n-i}$$

Answer 3

Use the very well known, textbook identity $$\sum_0^n \tbinom{n}{k} = (1+1)^n = 2^n $$ Can you show that your LHS is strictly smaller than the LHS of the above equation.

Answer 4

Hint: You don't need to calculate anything explicitly. What is the formula for $(1+x)^n$?