All Questions
5
questions with no upvoted or accepted answers
3
votes
2
answers
117
views
$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$
$$1\binom{20}1+2\binom{20}2+3\binom{20}3+\dots+19\binom{20}{19}+20\binom{20}{20}$$
I solved it by letting the sum be $S$, then adding the sum to itself but taking the terms from last to first and then ...
2
votes
2
answers
70
views
Question on changing the index of summation
$$b(a+b)^m = \sum_{j=0}^m \binom{m}{j}a^{m-j}b^{j+1}= \sum_{k=1}^m \binom{m}{k-1}a^{m+1-k}b^{k}+b^{m+1}$$
I believe $j = k-1$ though the book does say that.
This is related to proving the binomial ...
- 21
1
vote
0
answers
48
views
seperating two variables in a function with summation
I'm building a data analysis program that perform on big chunks of data, the issue I'm having is the speed of some operations; to be exact I have a function that takes two variables in this form : $$f(...
- 113
0
votes
0
answers
104
views
series based on $(1+x+x^2)^n$
Question : Let $a_r$ denote the following $$(1+x+x^2)^n=\sum_{r=0}^{2n}a_rx^r$$
then prove the following
$$\sum_{r=0}^{n}(-1)^r\binom n r a_r = \begin{cases}
0 & n \ne 3k \text{ for all ...
- 931
0
votes
0
answers
33
views
Solve for $x$: $\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$
Solve for $x$: $$\sum_{r=0}^5 {5\choose r} (-1)^rx^{5-r}3^r = 32$$
Looks like binomial theorem. So this would simplify to $(-x+3)^5=32$, and solving gives $x=2$. Is this correct?
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