Răspuns:
A = A1 * A2
Explicație pas cu pas:
A1 = 2+4+6+...+4046 = 2 (1+2+ ... + 2023) = 2 * [tex]\frac{2023 * 2024}{2}[/tex] = 2023 * 2024
am aplicat Suma Gauss 1+2+...+.n = n (n+1) / 2
A2 = [tex]\frac{1}{1 * 2} + \frac{1}{2*3} + ... + \frac{1}{2023 * 2024} = \frac{2-1}{1*2} +\frac{3-2}{2*3} +...+\frac{2024-2023}{2023*2024} = \frac{2}{1*2} -\frac{1}{1*2} + \frac{3}{2*3} -\frac{2}{2*3} +...+\frac{2024}{2023*2024} - \frac{2023}{2023 * 2024}[/tex]
simplificam pe fiecare fractie 2 cu 2, 3 cu 3, etc si vom obtine:
[tex]A 2 = 1 - \frac{1}{2} +\frac{1}{2} - \frac{1}{3} + ... - \frac{1}{2023} + \frac{1}{2023} - \frac{1}{2024} = 1 - \frac{1}{2024} =\frac{2024-1}{2024} = \frac{2023}{2024}[/tex]
( se observa ca termenii intermediari se reduc)
A = A1*A2 = [tex]A = A1 * A2 = 2023 * 2024 * \frac{2023}{2024} = 2023^{2}[/tex] care este patrat perfect
