Question

Aratati ca urmatoarele numere sunt patrate perfecte:
d)z=5+4*5+4*5^2+4*5^3+...+4*5^11

Answer 1

d)
z=5+4*5+4*5^2+4*5^3+.....+4*5^11
z=5+4*5^1+4*5^2+4*5^3+....+4*5^11
z=5+4*5*(5^0+5^1+5^2+5^3+....+5^10)
z=5+20*(5^0+5^1+5^2+5^3+...+5^10)
z=(20+5)*5^10
z=25*5^10
z=5^2*5^10
z=5^(2+10)
z=5^12
z=5^(6*2)
z=(5^6)^2⇒patrat perfect

Answer 2

z=5+4×5+4×5²+4×5³+....+4×$5^{11}$
z=5+4×5(1+4×5²+4×5³+....+4×$5^{10}$)
z=5+20(1+5+5²+5³+....+$5^{10}$)

1+5+5²+....+$5^{10}$=
[1+5]+[5²+5³]+...+$5^{9}$+$5^{10}$
=6+5²[5+1]+...+$5^{9}$[5+1]
=6+6·5²+...+6×$5^{9}$
=2×3[1+5²+...+$5^{9}$]⇒