Question

VA ROG
DAU COROANĂ
AJUTOR

Answer 1

13.

suma Gauss S =[n(n+1)]:2=

a) S=[49(49+1)]:2=1225

b)

2+4+6+8+...+198=2(1+2+3+....+99)=2×[(99×100)/2]=99×100=9900

c)

51+52+53+....+99=

(1+2+3.....+99) - (1+2+3+......+50) =

(1+2+3.....+99) =(99×100):2=4950

(1+2+3+......+50) =(50×51):2=1275

S=4950-1275=3675

d)

48+50+52+....+100=2(24+25+26+.....+50)

(24+25+26+.....+50)=(1+2+3+....+50)-(1+2+3+.....+23)

=(1+2+3+....+50)=(50×51):2=1275

=(1+2+3+.....+23)=(23×24):2=276

1275-276=999

dar 48+50+52+....+100=2(24+25+26+.....+50) deci 2(24+25+26+.....+50)=2×999=1998

14.

$a)2^{x}=32\\ \\ 2^{x}=2^{5} \\ \\ x=5$

$b) 3\cdot2^{2x+1} =3\cdot2\cdot 2^{2} \\ \\ 2^{2x+1} =2^{3} \\ \\ 2x+1=3\\ \\ 2x=3-1\\ \\ 2x=2\\ \\ x=1$

$c)5^{1}\cdot 5^{2}\cdot5^{3}.....\cdot5^{x}=5^{210} \\ \\ 5^{1+2+3+....+x}= 5^{210} \\ \\ 1+2+3+....+x=210\\ \\ 1+2+3+....+x=[x\cdot(x+1)]:2\\ \\ x(x+1)=210\cdot2\\ \\ x^{2}+x-420=0$

Δ=b²-4ac=1-4×(-420)=841

x₁=-15 si x₂=14

15.

$(3^{1}+3^{2} +3^{3}+....+3^{100}$