Prove that if you move straight down in Pascal's Triangle, visiting every other row, then the numbers are increasing.
Asked
Active
Viewed 135 times
0
-
1so you want to prove $\binom{n}{k}\leq \binom{n+1}{k}$? – Asinomás Feb 17 '17 at 00:34
-
Wouldn't it be (n+2 k+1) because it's every other row – JanoyCresva Feb 17 '17 at 00:37
-
I'm not sure, I'm confused how you would even derive that. – JanoyCresva Feb 17 '17 at 00:39
1 Answers
2
...prove that $\frac{n!}{2(n/2)!}$ for $2 | n$ is increasing as $n$ increases? Or down starting at arbitrary $k$ ans $n$? The latter is also fairly trivial:
$\binom{n+2}{k+1} = \binom{n+1}{k} + \binom{n+1}{k+1} = \bigg(\binom{n}{k-1} + \binom{n}{k}\bigg) + \binom{n+1}{k+1} > \binom{n}{k} $
dasaphro
- 324
-
-
Yes, so long as you have proved that $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$. This of course implies, more strongly, that every element of each row of Pascal's triangle is larger than every element of the previous row, which immediately proves your statement. – dasaphro Feb 17 '17 at 04:36