gamja
Active Member
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- Dec 14, 2022
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- HSC
- 2023
b)
i. Show that for all positive integers n,
^{n-1}+\left(1+x\right)^{n-2}+...+\left(1+x\right)^{2}+\left(1+x\right)+1\right]=\left(1+x\right)^{n}-1.)
ii. Hence show that for 1≤k≤n,
iii. Show that
\binom{n}{k+1}.)
iv. By differentiating both sides of the identity in (i), show that for 1≤k<n
\binom{n-2}{k-1}+\left(n-2\right)\binom{n-3}{k-1}+...+k\binom{k-1}{k-1}=k\binom{n}{k+1}.)
I'm basically screwed for all the questions since i'm stuck from the 1st... any hints? Thank you!
i. Show that for all positive integers n,
ii. Hence show that for 1≤k≤n,
iii. Show that
iv. By differentiating both sides of the identity in (i), show that for 1≤k<n
I'm basically screwed for all the questions since i'm stuck from the 1st... any hints? Thank you!
