Question

Aratati ca urmatoarele numere sunt patrate perfecte , dupa care calculati cu √.
a) x = 6 + 12 + 18 + ... + 288
b) x = 4 + 8 + 12 + ... + 196
c)x=1+3+5+.....+101
d)x=1+3+5+...+2011
e)x=1203+2+4+6+......2404

VA ROG URGENT

+ daca aveti timp :
x=1+2+3+...49
x=1+3+5+....2013
x=1+3+5....+81
x=2+4+6+....2012 + 1007

Answer 1

a) x = 6 + 12 + 18 + ... + 288=6*(1+2+3+4...+48)=6*48*49/2=7056

b) x = 4 + 8 + 12 + ... + 196=4*(1+2+3+..+49)=4*49*50/2=4900

c)x=1+3+5+.....+101=51²=2050 ( formula  pt. adunarea numerol impare consecutive est n² )
a1=2n-1=2*1-1=1
a2=2*2-1=3
2*n-1=101
2n=102  :2
n=51

d)x=1+3+5+...+2011= 1006²=1 012 036
2*n-1=2011
2n=2012  :2
n=1006

e)x=1203+2+4+6+......2404=1203+2*(1+2+3+...+1202)=
 =1203+ 2*1202*1203/2=1203+ 1202*1203= 1203*(1+1202)= 1203²=
=1 447 209


x=1+2+3+...49= 49*50/2=49*25=1225

x=1+3+5+....2013=1007²=1 014 049
2n-1=2013
2n=2014
n=1007

x=1+3+5....+81=41²=1681
n=41

x=2+4+6+....2012 + 1007 = 2*(1+2+...1006)+1007= 2*1007*1006/2+ 1007=
=1007*(1006+1)=1007²= 1 014 049