Question

determinati n:

a)5 la puterea 1+2+3+...+2014=5 la puterea n+1007

b)7 la puterea 2+4+6+....+100=7 la puterea n(n+1)

Answer 1

1) 5 ^ (1+2+3+....+2014)=5 ^ (n+1007)

1+2+3+...+2014=n+1007

(1+2014)*2014/2= n+1007

2015*1007=n+1007
2 029 105=n+1007

n=2 029 105 -1007
n=2 028 098

2)7 ^(2+4+6+...+100)=7 ^n(n+1)

2+4+6+...+100=n(n+1)

2(1+2+3...+50)=n(n+1)

2*(1+50)*50/2= n(n+1)
(50+1)*50=n(n+1)

n=50
n+1=50+1=51

Answer 2

a)5 la puterea 1+2+3+...+2014=5 la puterea n+1007

1+2+3+...+2014 = n(n+1)/2
=2014*2015/2

2014*2015/2=n+1007
2014*2015=2(n+1007)
2n+2014=2014*2015
2n=2014*2015-2014
2n=2014(2015-1)
2n=2014*2014
n=2014*2014/2
n=1007*2014


b)7 la puterea 2+4+6+....+100=7 la puterea n(n+1)
2+4+6+...+100=
2(1+2+3+4+..3.+50)=
2*(50*51/2)= 2 cu 2 se simplifica
50*51

50*51=n(n+1)= > clar ca n=50