If $T(n) = 1 + \sum_{k=0}^{n-1} T(k),$ then $T(n) = 2^n.$
105882
Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$
122534
Show that $\sum (-1)^{n+1}/n$ = $k$ can be expanded to a sum with $ 1 - 1/2 + 1/3 + 1/5 - 1/4 +1/7 + 1/9 - 1/6 + ....$ which is $3k/2$
148442
Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
37771
Sum of a power series $n x^n$
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
52324
Tight bound for $T(n) = T(n^{1/2}) + 1$
131400
Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
58144
Solve the recurrence relation:$ T(n) = \sqrt{n} T \left(\sqrt n \right) + n$
107071
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
36436
Proving $\sum_{k=1}^n k k!=(n+1)!-1$