Question

Va rog ajutați-mă calculați sumele :
1+2+3+....+100
1+3+5+.....+99
100+101+102+....+200
2+4+6+.....+100
3+6+9+....+300
101+103+...+199
Calculati:
A 2 la 10*25la2*7la10=
B 2la5*5la7*3la10=

Answer 1

1+2+3+....+100 = suma Gauss
S=[n(n+1)]:2=(100×101):2=5050

1+3+5+.....+99 =
progresie aritmetica :a1=1 , an=99 si r=2 (ratia)
Formula an=1+(n-1)×r⇒
99=1+(n-1)×2(=)98=2×(n-1)⇒49=n-1⇒n=50⇒
90 este al 50 -lea termen al sirului =>
Formula suma primilor n termeni ai unui sir este  (a1+an)×n/2 ⇒
Suma termenilor sirului este (1+99)×50/2=100×25=2500

S=100+ 101+102+103+......+200⇒S=(101+200)×100/2⇒S=301×50+100⇒S=15050+100=15150


2+4+6+.....+100=2×(1+2+3+....+100)=2×5050=10100

3+6+9+....+300=3×(1+2+3+....+100)=3×5050=15150

101+103+...+199=progresie aritmetica cu ratia 2
nr de termeni = N
N=(an - an):r+1
N=(199-101):2+1
N=98÷2+1
N=49+1
N=50S=[(a1+an)×nt]:2
S=[(101+199)×50]:2
S=(300×50):2
S=15000:2
S=7500

A) x = 2¹⁰ × (5²)²× 7¹⁰ = 2¹⁰× 5⁴ × 7¹⁰ = (2 × 5)⁴ × 2⁶× 7¹⁰ = 10⁴ × 2⁶ × 7¹⁰

B) y = 2⁵ × 5⁷ × 3¹⁰ = (2 × 5)⁵ × 5² × 3¹⁰ = 10⁵ × 5² × 3¹⁰