Po prostu kilkukrotnie stosujemy wzór na sumę ciągu geometrycznego: x + 2x^2 + 3x^3 + 4x^4 + .... + n*x^n = (x + ... + x^n) + (x^2 + ... + x^n) + (x^3 + ... + x^n) + ... + x^n x ≠ 1 Sn = a₁(1 - q^n)/(1 - q) = x(1 - x^{n})/(1 - x) + x^2 (1 - x^{n - 1})/(1 - x) + x^3 (1 - x^{n - 2})/(1 - x) + ... + x^n (1 - x^1)/(1 - x) = [x(1 - x^{n}) + x^2 (1 - x^{n - 1}) + x^3 (1 - x^{n - 2}) + ... + x^n (1 - x^1)]/(1 - x) = [x - x^{n + 1} + x^2 - x^{n + 1}) + x^3 - x^{n + 1}) + ... + x^n - x^{n + 1}]/(1 - x) = [(x + x^2 + x^3 + ... + x^n) - nx^{n + 1}]/(1 - x) = [x(1 - x^{n})/(1 - x) - nx^{n + 1}]/(1 - x) = [x(1 - x^{n}) - (1 - x)*nx^{n + 1}]/(1 - x)² = [x - x^{n + 1} - nx^{n + 1} + nx^{n + 2}]/(1 - x)² = [x - (n + 1)x^{n + 1} + nx^{n + 2}]/(1 - x)² x = 1 x + 2x^2 + 3x^3 + 4x^4 + .... + n*x^n = 1 + ... + n = (n + 1)n/2 jak masz pytania to pisz na pw
