Zadanie 4.63
Wyznacz n, wiedząc, że n ∊ N
a)
n+{n\choose 2}=15
n+\frac{n!}{(n-2)!\cdot 2!}=15
n+\frac{(n-2)(n-1)n}{(n-2)!\cdot 2}=15
n+\frac{(n-1)n}{2}=15 \ |*2
2n+n^2-n=30
n^2+n-30=0
a=1, b=1, c=-30
\Delta=b^2-4ac=1-4\cdot 1\cdot (-30)=121
\sqrt\Delta=11
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-1-11}{2}=-6\not\in \mathbb N odrzucamy
x_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-1+11}{2}=5
n=5
b)
{n\choose n-1}+{2n\choose 2n-2}=18 , n\in \mathbb N
\frac{n!}{[n-(n-1)]!\cdot (n-1)!)!}+\frac{(2n)!}{[2n-(2n-2)]!\cdot (2n-2)!}=18
\frac{(n-1)!\cdot n}{1!\cdot (n-1)!}+\frac{(2n-2)!\cdot (2n-1)\cdot 2n}{2!\cdot (2n-2)!}=18
n+\frac{\not2^1n(2n-1)}{\not2^1}=18
n+n(2n-1)=18
n+2n^2-n=18
2n^2=18 \ |:2
n^2=9 …-3\not\in N odrzucamy
n=3