Question

Demonstrati egalitatea,folosind metoda inductiei matematice:

1²-2²+3²-4²+...+(-1)ⁿ⁻¹ * n²= (-1)ⁿ⁺¹ * n(n+1)/2

Answer 1

1²-2²+3²-4²+...+(-1)ⁿ⁻¹ * n²= (-1)ⁿ⁺¹ * n(n+1)/2

1. Verificare
P₁:  (-1)¹⁻¹ * 1²= (-1)¹⁺¹ * 1(1+1)/2
=>  1=2/2   adevarat
P₂: 1+(-1)²⁻¹ * 2²= (-1)²⁺¹ * 2(2+1)/2
=> 1-4=-6/2 <=> -3=-3 adevarat

2.
Pk->Pk₊₁
Pk: $1^2-2^2+3^2-4^2+...+(-1)^{k-1} * k^2=\frac{(-1)^{k+1} *k(k+1)}{2}$  pp ca e adevarat

Pk₊₁: $1^2-2^2+3^2-4^2+...+(-1)^{k-1} * k^2+(-1)^{k+1-1} * (k+1)^2= \frac{(-1)^{k+1+1} *( k+1)(k+1+1)}{2}$

=> $\frac{(-1)^{k+1} *k(k+1)/}{2}+(-1)^{k+1-1} * (k+1)^2= \frac{(-1)^{k+1+1} *( k+1)(k+1+1)}{2}$

=> $\frac{(-1)^{k+1} *k(k+1)}{2}+(-1)^k * (k+1)^2= \frac{(-1)^{k+2} *( k+1)(k+2)}{2}$               |*2

$\-(-1)^k *k(k+1)+2*(-1)^k * (k+1)^2= (-1)^k *( k+1)(k+2)$     |    :(-1)^k

-k(k+1)+2(k+1)²=(k+1)(k+2)
-k²-k+2k²+4k+2=k²+3k+2
k²+3k+2=k²+3k+2 adevarat=>
=>Pk este adevarat