Question

S=f(0)+f(1)+...+f(2011)
f : R --->R, f(x)=-2x+1
S=?
Daca puteți, pe pasi!

Answer 1

f : R ---> R
f(x)= -2x+1
S=f(0)+ f(1)+...+f(2011)
f(0)= -2*0+1 = 1
f(1) = -2*1+1 = -1
f(2) = -2*2+1 = -3
f(3) = -2*3+1 = -5
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.
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f(2010) = -2*2010+1 = -4020+1 = -4019
f(2011) = -2*2011+1 = -4022+1 = -4021
 Inlocuim f(0) cu 1 , f(1) cu -1, etc. si obtinem:
S= 1 + (-1) + (-3) +(-5)+...+(-4019) + (-4021) 
S= 1 -1 - 3 - 5 -....-4019 - 4021
S= 1 - (1+3+5+...+4019+4021)
 1+3+5+......+4019+4021=2011*2011=4044121 deoarece aplicam formula : 1+3+5+...+2n-1 = n*n
Deci S=1-4044121 = -4044120