Question

Calculati termenii x1 , x2 , .... x10 ....xn si suma primilor 100 de termeni

a ) 1,4,7 ...
b) 2,7,12 ...
c ) 4,12,20...
d ) 10,15,20
e) 31,35,30 ...

Answer 1

a) 1,4,7....
x₁=1
x₂=1+3=4
x₁₀=1+3x9=28
xn=1+3(n-1)
x₁+x₂+x₃+....+x₁₀₀=1+4+7+....298=1+(1+3)+(1+6)+....(1+297)=100*1+(3+6+..297)=
=100+3(1+2+....99)=100+3*99*(99+1)/2=100+3*99*50=14950

b) 2,7,12....
x₁=2
x₂=2+5=7
x₁₀=2+5x9=47
xn=2+5(n-1)
x₁+x₂+x₃+....+x₁₀₀=2+7+12+....497=2+(2+5)+(2+10)+....(2+495)=100*2+(5+10+..495)=200+5(1+2+....99)=200+5*99*(99+1)/2=200+5*99*50=24950

c) 4,12,20....
x₁=4
x₂=4+8=12
x₁₀=4+8x9=76
xn=4+8(n-1)
x₁+x₂+x₃+....+x₁₀₀=4+12+20+....796=4+(4+8)+(4+16)+....(4+792)=100*4+(8+16+..99)=400+8(1+2+....99)=400+8*99*(99+1)/2=400+8*99*50=40000

d) 10,15,20....
x₁=10
x₂=10+5=15
x₁₀=10+5x9=55
xn=10+5(n-1)
x₁+x₂+x₃+....+x₁₀₀=10+15+20+....505=10+(10+5)+(10+10)+....(10+495)=100*10+(5+10+..495)=1000+5(1+2+....99)=1000+5*99*(99+1)/2=1000+5*99*50=25750

d) 31,35,39....
x₁=31
x₂=31+4=35
x₁₀=31+4x9=67
xn=31+4(n-1)
x₁+x₂+x₃+....+x₁₀₀=31+35+39+....427=31+(31+4)+(31+8)+....(31+396)=100*31+(4+8+..396)=
=3100+4(1+2+....99)=3100+4*99*(99+1)/2=3100+4*99*50=22900