# Find the sum of $\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$

Find the sum of $\binom{100}1 + 2\binom{100}2 + 4\binom{100}3 +8\binom{100}4+\dots+2^{99}\binom{100}{100}$

How you guys work on with this question? With the geometric progression? Combination? Or anyother way to calculate?

• Have you tried expanding $(1 + 2)^{100}$ using the binomial theorem? – Arthur Mar 3 '14 at 16:04
• There was a misplaced parenthesis in the original question. Please check that the latexification was done right. – Daniel R Mar 3 '14 at 16:13
• Thx for help in edit my question.... – user132564 Mar 3 '14 at 16:25
• It looks like you have some extra parentheses. – Ross Millikan Mar 3 '14 at 16:33

$$\sum_{r=1}^{100}2^{r-1}\binom{100}r=\frac12\sum_{r=1}^{100}2^r\binom{100}r=\frac12\left[(1+2)^{100}-1\right]$$