Question

Calculati produsul x*y,unde
x=-3/4*-3/4 la puterea 2 *...........-3/4 la puterea 10
y=-4/3*-4/3 la puterea 2*............-4/3 la puterea 10

Answer 1

x = -3/4 [1+3/4 + (3/4)² +(3/4)³ +.........+(3/4)^9] = -3/4 ·S1
 y = -4/3[1+4/3 +(4/3)² +(4/3)³ +.......+(4/3)^9] = -4/3 ·S2
x·y = S1 ·S2
S1= 1+3/4 + (3/4)² +(3/4)³ +........+(3/4)^9      3/4 ·S1 = 3/4 + (3/4)² +(3/4)³ +....+(3/4)^10 ⇒ S1 - 3/4 ·S1 = 1/4 ·S1 = 1 - (3/4)^10 ⇒ S1 = 4[1-(3/4)^10]
S2 = 1 + 4/3 +(4/3)² + ....+(4/3)^9    4/3 ·S2 = 4/3 + (4/3)² +(4/3)³ +...........+(4/3)^10
4/3 ·S2 - S2 = 1/3 ·S2 = (4/3)^10 - 1 ⇒ S2 = 3[(4/3)^10 - 1]
x·y = S1 ·S2 = 12{ [1-(3/4)^10]·[(4/3)^10 - 1]=12[ (4/3)^10 - 1 -1 +(3/4)^10] =
= 12·[(4/3)^5 - (3/4)^5]²