Basics of Induction
Prove the following using induction: 1. $\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}$. 2. $\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}$. 3. $\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2$. 4....
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Prove the following : 1. $$\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k {n-k \choose k} \cdot 2^{n-2k} = n + 1$$ 2. $$\sum_{k=0}^n {2k \choose k}{2n-2k \choose n-k} = 4^n$$ 3. $$\sum_{k=0}^{n} 2^k {n \choose...
View ArticleSummations and Combinations
Prove the following : 1. $$\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k {n-k \choose k} \cdot 2^{n-2k} = n + 1$$ 2. $$\sum_{k=0}^n {2k \choose k}{2n-2k \choose n-k} = 4^n$$ 3. $$\sum_{k=0}^{n} 2^k {n \choose...
View ArticleBasics of Induction
Prove the following using induction: 1. $\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}$. 2. $\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}$. 3. $\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2$. 4....
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