Question

IMI TREBUIE URGENT PAINE MAINE DAU CORONITA CELUI MAI BUN RASPUNS SI MULTE PUNCTE
1. COMPARATI NR: A=4 LA PUTEREA 1+3+5+.......+31 SI B=(3+6+(+:..+33+36) LA PUTEREA 64
2.Aratati ca nr 5 la puterea 344 se poate scrie ca suma a trei cuburi perfecte

3.Se considera sirul: 10,13,16,19,...
a)Calculati al 2016 lea termen
b) Calculati suma primilor 100 termeni

4.Determinati nr de forma abc care impartite la bc dau catul si restul egal cu 18.

Answer 1

1. 1+3+5+...+31=(1+0)+(1+2)+(1+4)+(1+6)+....+(1+30)= (avem 16 termeni)
=16+(0+2+4+...+30)=16+2(0+1+2+...+15)=16+2·15·16/2=16+15·16=16+240=
=256
a=4^256=(4^4)^64=16^64
3+6+9+...+36=3(1+2+3...+12)=3·12·13/2=9·13=117
b=117^64
16^64<117^64
A<B
2.
5^344=5^(114·3+2)=25·5^(3·114)=(1+8+16)·(5^114)³
trebuie sa obtinem in paranteza o suma de cuburi, iar 16 nu e cub
=25·5³·(5^113)³=3125·(5^113)³=
 ma mai gandesc
3. se observa ca
Tk=10+3(k-1)
T1=10+3·0
T2=10+3·1
....
T2016=10+3(2016-1)=10+3·2015=10+6045=6055
Suma= 100·10+3·(suma de la 0 la 99)=1000+3·99·100/2=1000+3·99·50=
=1000+14850=15850

4. abc, bc nr in baza 10, a,b,c cifre, a,b≠0
abc:bc=18 rest 18, bc>18
abc=18bc+18=18(bc+1)
abc=100a+bc=18bc+18
100a=17bc+18
100a=17(10b+c)+18
100a=170b+(17c+18)
170b divizibil cu 10⇒(17c+18)div cu 10⇒c=6
100a=170b+120
10a=17b+12⇒(17b+12) div cu 10⇒b=4
10a=80, a=8
numarul abc = 846