- How to calculate the sum of this sequence ?
$$\binom{31}{1}2^{29} + \binom{31}{3}2^{27} + \binom{31}{5}2^{25} + \binom{31}{7}2^{23} + \binom{31}{9}2^{21} + \binom{31}{11}2^{19} + \binom{31}{13}2^{17} + \binom{31}{15}2^{15} + \binom{31}{17}2^{13} + \binom{31}{19}2^{11} + \binom{31}{21}2^9 + \binom{31}{23}2^7 + \binom{31}{25}2^5 + \binom{31}{27}2^3 + \binom{31}{29}2^1$$
- How to calculate the sum of this sequence
$\binom{10}{1}2^{9} + \binom{10}{2}2^{8} + \binom{10}{3}2^{7} + ...\binom{10}{10}2^{0}$
Is there any approach, how to calculate the sum ?
For the first question:
Using the Binomial Theorem, we have $$(2+1)^{31} = \dbinom{31}{0}2^{31}+\dbinom{31}{1}2^{30}+\dbinom{31}{2}2^{29}+\cdots+\dbinom{31}{29}2^2+\dbinom{31}{30}2^1+\dbinom{31}{31}2^0$$ $$(2-1)^{31} = \dbinom{31}{0}2^{31}-\dbinom{31}{1}2^{30}+\dbinom{31}{2}2^{29}-\cdots-\dbinom{31}{29}2^2+\dbinom{31}{30}2^1-\dbinom{31}{31}2^0$$
By subtracting the second equation from the first, we get $$3^{31}-1 = 2\left[\dbinom{31}{1}2^{30}+\dbinom{31}{3}2^{28}+\cdots+\dbinom{31}{29}2^2+\dbinom{31}{31}2^0 \right]$$
Can you manipulate this to get an expression for $\dbinom{31}{1}2^{29}+\dbinom{31}{3}2^{27}+\cdots+\dbinom{31}{29}2^1$?
For the second question:
Using the Binomial Theorem, we have $$(2+1)^{10} = \dbinom{10}{0}2^{10}+\dbinom{10}{1}2^9+\dbinom{10}{2}2^8+\dbinom{10}{3}2^7+\cdots+\dbinom{10}{10}2^0$$
Can you manipulate this to get an expression for $\dbinom{10}{1}2^9+\dbinom{10}{2}2^8+\dbinom{10}{3}2^7+\cdots+\dbinom{10}{10}2^0$?