Question

use mathematical induction to prove the following formula for positive integer n.

Answer 1

@waterineyes

Answer 2

idk

Answer 3

its true for n=1 because
\[\left(\begin{matrix}3 \\ 3\end{matrix}\right)=\left(\begin{matrix}4 \\ 4\end{matrix}\right)\]

Answer 4

now suppose that statement holds for \(n\) then we most prove its true for \(n+1\)

Answer 5

so we have
\[\left(\begin{matrix}3 \\3\end{matrix}\right)+\left(\begin{matrix}4 \\3\end{matrix}\right)+...+\left(\begin{matrix}n+2 \\3\end{matrix}\right)=\left(\begin{matrix}n+3 \\4\end{matrix}\right)\]

and we want to prove

\[\left(\begin{matrix}3 \\3\end{matrix}\right)+\left(\begin{matrix}4 \\3\end{matrix}\right)+...+\left(\begin{matrix}n+3 \\3\end{matrix}\right)=\left(\begin{matrix}n+4 \\4\end{matrix}\right)\]

Answer 6

\[\left(\begin{matrix}3 \\3\end{matrix}\right)+\left(\begin{matrix}4 \\3\end{matrix}\right)+...+\left(\begin{matrix}n+3 \\3\end{matrix}\right)=\left(\begin{matrix}3 \\3\end{matrix}\right)+\left(\begin{matrix}4 \\3\end{matrix}\right)+...+\left(\begin{matrix}n+2 \\3\end{matrix}\right)+\left(\begin{matrix}n+3 \\3\end{matrix}\right)\\=\left(\begin{matrix}n+3 \\4\end{matrix}\right)+\left(\begin{matrix}n+3 \\3\end{matrix}\right)\]

Answer 7

so we just need to prove
\[\left(\begin{matrix}n+3 \\4\end{matrix}\right)+\left(\begin{matrix}n+2 \\3\end{matrix}\right)=\left(\begin{matrix}n+4 \\4\end{matrix}\right)\]

Answer 8

am i clear?

Answer 9

yes
what next ?

Answer 10

sorry my last reply is

\[\left(\begin{matrix}n+3 \\4\end{matrix}\right)+\left(\begin{matrix}n+3 \\3\end{matrix}\right)=\left(\begin{matrix}n+4 \\4\end{matrix}\right) \]

Answer 11

now just note that...

\[\left(\begin{matrix}n+3 \\4\end{matrix}\right)+\left(\begin{matrix}n+3 \\3\end{matrix}\right)=\frac{(n+3)!}{(n-1)! \ 4!}+\frac{(n+3)!}{n! \ 3!}=\frac{n(n+3)!+4(n+3)!}{n! \ 4!}=\frac{(n+4)!}{n! \ 4!}\\=\left(\begin{matrix}n+4 \\4\end{matrix}\right) \]

Answer 12

so the proof is complete and that statement holds for all \( n \in N\)

Answer 13

thanks for your help:)

Answer 14

yw........:)