Potencia
Páginas: 2 (469 palabras)
Publicado: 8 de agosto de 2012
(n0)(x-0)n-(n1) (x-1)n+ (n2) (x-2)n-…± (nn-1) (x-n+1)n ± (nn) (x-n)n=n!
Si p(n)=1
(10)(x-0)1-(11) (x-1)1=1!
→ (1)(x)-(1)(x-1)=1!
→x-x+1=1!
→ 1=1
Luego suponemos que p(n) [(n0)(x-0)n-(n1) (x-1)n+ (n2) (x-2)n-…± (nn-1) (x-n+1)n ± (nn) (x-n)n=n!] es verdadero
Entonces para p(n+1)(n+10)(x-0)n+1-(n+11) (x-1)n+1+…± (n+1n) (x-n)n+1 ± (n+1n+1) (x-n+1)n+1=(n+1)!
Xn+1-(n+11) (x-1)n+1+...± (n+1n) (x-n)n+1 ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0)+(n1)] (x-1)n(x-1)}+...± {[(nn-1)+ (nn)] (x-n)n(x-n)} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0) (x-1)n(x-1)+ (n1) (x-1)n(x-1)]}+...± {[(nn-1) (x-n)n(x-n)+ (nn)(x-n)n(x-n)]} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0) (x-1)nx -(n0) (x-1)n]+ [(n1) (x-1)nx-(n1) (x-1)n]}+...± {[(nn-1) (x-n)nx-n(nn-1) (x-n)n]+ [(nn) (x-n)nx-n(nn)(x-n)n]} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(x-1)nx -(x-1)n]+ [(n1) (x-1)nx-(n1) (x-1)n]}+...± {[(nn-1) (x-n)nx-n(nn-1) (x-n)n]+ [(x-n)nx-n(x-n)n]}±(x-n+1)n+1=(n+1)!
XnX-[(x-1)nx -(x-1)n+ (n1) (x-1)nx-(n1) (x-1)n]+...± [(nn-1) (x-n)nx-n(nn-1) (x-n)n+ (x-n)nx-n(x-n)n] ±(x-n+1)n+1=(n+1)!
XnX-(x-1)nx +(x-1)n- (n1)(x-1)nx+(n1) (x-1)n+...± [(nn-1) (x-n)nx-n(nn-1) (x-n)n+ (x-n)nx-n(x-n)n] ±(x-n+1)n+1=(n+1)!
x[Xn - (n1) (x-1)n +...± (x-n)n]+ x[-(x-1)n+… ±(nn-1)(x-n)n ]+[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1) (x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
x[n!]+ x[-(x-1)n+… ±(nn-1) (x-n)n ]+[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1)(x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
X[n!+(-(x-1)n+…±(nn-1) (x-n)n)] +[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1) (x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
Si p(n)=1
(10)(x-0)1-(11) (x-1)1=1!
→ (1)(x)-(1)(x-1)=1!
→x-x+1=1!
→ 1=1
Luego suponemos que p(n) [(n0)(x-0)n-(n1) (x-1)n+ (n2) (x-2)n-…± (nn-1) (x-n+1)n ± (nn) (x-n)n=n!] es verdadero
Entonces para p(n+1)(n+10)(x-0)n+1-(n+11) (x-1)n+1+…± (n+1n) (x-n)n+1 ± (n+1n+1) (x-n+1)n+1=(n+1)!
Xn+1-(n+11) (x-1)n+1+...± (n+1n) (x-n)n+1 ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0)+(n1)] (x-1)n(x-1)}+...± {[(nn-1)+ (nn)] (x-n)n(x-n)} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0) (x-1)n(x-1)+ (n1) (x-1)n(x-1)]}+...± {[(nn-1) (x-n)n(x-n)+ (nn)(x-n)n(x-n)]} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(n0) (x-1)nx -(n0) (x-1)n]+ [(n1) (x-1)nx-(n1) (x-1)n]}+...± {[(nn-1) (x-n)nx-n(nn-1) (x-n)n]+ [(nn) (x-n)nx-n(nn)(x-n)n]} ±(x-n+1)n+1=(n+1)!
Xn+1-{[(x-1)nx -(x-1)n]+ [(n1) (x-1)nx-(n1) (x-1)n]}+...± {[(nn-1) (x-n)nx-n(nn-1) (x-n)n]+ [(x-n)nx-n(x-n)n]}±(x-n+1)n+1=(n+1)!
XnX-[(x-1)nx -(x-1)n+ (n1) (x-1)nx-(n1) (x-1)n]+...± [(nn-1) (x-n)nx-n(nn-1) (x-n)n+ (x-n)nx-n(x-n)n] ±(x-n+1)n+1=(n+1)!
XnX-(x-1)nx +(x-1)n- (n1)(x-1)nx+(n1) (x-1)n+...± [(nn-1) (x-n)nx-n(nn-1) (x-n)n+ (x-n)nx-n(x-n)n] ±(x-n+1)n+1=(n+1)!
x[Xn - (n1) (x-1)n +...± (x-n)n]+ x[-(x-1)n+… ±(nn-1)(x-n)n ]+[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1) (x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
x[n!]+ x[-(x-1)n+… ±(nn-1) (x-n)n ]+[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1)(x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
X[n!+(-(x-1)n+…±(nn-1) (x-n)n)] +[(x-1)n -… ±n(nn-1) (x-n)n ]+[(n1) (x-1)n -…± n(x-n)n] ±(x-n+1)n+1 =(n+1)!
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