Could someone please help me with this. Use the pascal sequence property nCr + nC(r+1) = (n+1)C(r+1) to explain why F3 + F4 = F5 and F4 + F5 = F6.
Given that Fn = nC0 +(n-1)C1 +...+ 1C(n-1) + 0Cn.
What I tried was to expand out F3, F4, F5 and F6 and then use the pascal sequence on terms which had r higher than 0.
For example. F6 = 6C0 +5C1 +4C2 +3C3 + 2C4 + 1C5 + 0C6.
By the pascal sequence 5C1 = 4C0 + 4C1 and 4C2 = 3C1 + 3C2 etc. But what should I do with terms where n is larger than r like 2C4 and 1C5??? I ditched them because they are meaningless. Fair enough?
Thanks in advance for any assistance.
Now group the terms like this:
Now use Pascal's Identity on the grouped terms:
Note 2 things:
1) , so we can add it to the sum without affecting the equality.
2) , so we can replace by
But by definition,