Question

Rezolvati prin suma lui Gauss:
a)1+2+3+...+2017 =
b)7+8+9+...+50=
c)31+32+...+50=
d)1+2+3+...+30=
e)1+3+5+...+29=
f)2+4+6+...+30=

Urgent! Ofer coronita celui mai bun raspuns!

Answer 1

a)

1 + 2 + 3 +.......+ 2 017 = 2017 × ( 1 + 2017 ) / 2 = 2017 × 2018 / 2 =

= 2017 × 1009 = 2 035 153

b) 7 + 8 + 9 + ..........+ 50 = 44 × 57 / 2 = 22 × 57 = 1 254

50 - 6 = 44 termeni ai sumei

c) 31 + 32 + ... + 50 = 20 ×( 31 + 50 ) / 2 = 10 × 81 = 810

50 - 30 =20 termeni ai sumei

d) 1 + 2 + 3 + ... + 30 = 30 × ( 1 + 30 ) / 2 = 15 × 31 = 465

e) 1 + 3 + 5 + ... + 29 = 1 + ( 1 + 2) + ( 1 + 4 ) + ..........+ ( 1 + 28 ) =

= 1 + 1 + 1 + .......+ 1 + 2 ( 1 + 2 + 3 + 4 + ........+ 14 ) =

= 15 × 1 + 2 × ( 14 × 15 ) / 2 = 15 + 210 = 225

f) 2 + 4 + 6 + .. . + 30 = 2 ( 1 + 2 + 3 +.........+ 15 ) = 2 × 15 × 16 / 2 = 15 × 16 = 240

Answer 2

exercitiile a) - d) se rezolva cu formula...

,, 1+2+3+...+n = n*(n+1) / 2 "

a)1+2+3+...+2017 = 2017*(2017+1)/2 = 2017*1009 = 2035153

b)7+8+9+...+50 = 7+(1+1*7)+(1+2*7)+ ... +(+7*7)= 1*8+7*(1+2+...+7)=8+7*8/2=36

c)31+32+...+50 = (31) + (31+1*1)+(31+1*2)+...+(31+1*19) =

= 31*20 + 1*(1+2+...+19)=620+19*20/2= 620+190 = 810

d)1+2+3+...+30 = 30*31/2 = 15*32 = 480

se foloseste formula...

,, 1+3+5+...+2n-1 = n² "

e)1+3+5+...+29 = 15²= 225 (pentru ca 2n-1=29 deci n = (29+1)/2 = 15 !)

se foloseste formula...

,, 2+4+6+...+2n = n*(n+1) "

f)2+4+6+...+30= 15*16 = 240 (pentru ca 2n = 30 deci n = 15 !)