Question

Calculati suma S= f(1)+f(2)+....+f(100) .

Answer 1

f(1)=8-7=1
f(2)=16-7=9
f(3)=24-7=17
f(4)=32-7=25
....f(n)=f(n-1)+8
=> progresie aritmetica cu f(1)=1
ratia r=8
an=f(100)=800-7=793
S₁₀₀=(a1+a₁₀₀)*100/2
S=(1+793)*50
S=39 700

Answer 2

$\\ \boxed{\text{Rezolvarea clasica:}}$

$f(x) = 8x-7\\ \\ \sum\limits_{x=1}^{100}f(x) = \sum\limits_{x=1}^{100}(8x-7) = \sum\limits_{x=1}^{100}8x - \sum\limits_{x=1}^{100}7 = 8\cdot \sum\limits_{x=1}^{100}x-\sum\limits_{x=1}^{100}7 = \\ \\ = 8\cdot (1+2+3+...+100)-(7+7+7+\underset{\text{de 100 ori}}{\underbrace{...}}+7) = \\ \\ = 8\cdot \dfrac{100\cdot (100+1)}{2}-7\cdot 100 = \\ \\ = 4\cdot 100\cdot 101-7\cdot 100 = \\ \\ = 100\cdot(4\cdot 101-7) = \\ \\ = 100\cdot (404-7) = \\ \\ = 100\cdot 397 = \\ \\ = 39700$