If Bx denotes the kth term in the expression of (1+y)ⁿ in descending power of...

If Bx denotes the kth term in the expression of (1+y)ⁿ in descending power of n∈N. Show that k(k+1)Bₓ₊₂=(n-k+1)(n-k)y²Bₓ?

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Answers (1)

busari
1 month ago
The binomial theorem gives us (a+b)n=∑vk=0n!k!(n−k)!akbn−k

So, (1+y)n=(y+1)n=∑vk=0n!k!(n−k)!yk1n−k

Hence, the kth term of (1+y)n is Bk=n!k!(n−k)!yk

So, assuming k+2≤n ,

Bk+2=n!(k+2)!(n−k−2)!yk+2=n!k!(n−k)!yk(n−k)(n−k−1)(k+1)(k+2)y2
=Bk(n−k)(n−k−1)(k+1)(k+2)y2
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