Question

Calculati sumele:
a) 2+4+6+...+98+100=
b)3+6+9+...+180=
c)4+8+12+...360=
d)5+10+15+...+625=
e)1+3+5+...+201=
f)1+4+7+...+301=
g)2+7+12+...+177=
h)3+7+11+...+199=
i)1+6+11+...+2006+2011=
j)1+4+7+...+2011+2014=
k)4+9+14+...+2009+2014=

Dau coroana si 10p.

Answer 1

a)
2+4+6+......+98+100=... este de forma 2+4+6+........+2n=n*(n+1);
2n=100 => n=100:2 => n=50
S=50*51=2550
b)
3+6+9+.....+180=.... este de forma 3+6+9+.....+3n=n*(n+1)/2*3;
3n=180 => n=180:3 => n=60
S=60*61/2*3=5490
c)
4+8+12+....+360=... este de forma 4+8+12+...+4n=n*(n+1)*2;
4n=360 => n=360:4 => n=90
S=90*91*2=16380
d)
5+10+15+....+625=.... este de forma 5+10+15+....+5n=n*(n+1)/2*5;
5n=625 => n=625:5 => n=125
S=125*126/2*5=39375
e)
1+3+5+.....+201=... este de forma 1+3+5+...+2n-1=n*n=n²;
2n-1=201 => 2n=201+1 => 2n=202 => n=202:2 => n=101
S=101*101=101²=10201
f)
1+4+7+....+301=... este de forma 1+4+7+...+3n-2=n*(3n-1):2
3n-2=301 => 3n=301+2 => 3n=303 => n=303:3 => n=101
S=101*(303-1):2=101*302:2=30502:2=15251
g)
2+7+12+.....+177=... este de forma 2+7+12+....+5n-3=(n:2)(5n-1)
5n-3=177 => 5n=177+3 => 5n=180 => n=180:5 => n=36
S=(36:2)(180-1)=18*179=3222
h)
3+7+11+...+199=... este de forma 3+7+11+.....+4n-1=n(2n+1)
4n-1=199 => 4n=199+1 => 4n=200 => n=200:4 => n=50
S=50*(100+1)=50*101=5050
i)
1+6+11+.....+2006+2011=... este de forma 1+6+11+...+5n-4=n(5n-3):2
5n-4=2011 => 5n=2011+4 => 5n=2015 => n=2015:5 => n=403
S=403*(2015-3):2=403*2012:2=810836:2=405418
j)
1+4+7+....+2011+2014=... este de forma 1+4+7+....+3n-2=n*(3n-1):2
3n-2=2014 => 3n=2014+2 => 3n=2016 => n=2016:3 => n=672
S=672*(2016-1):2=672*2015:2=1354080:2=677040

Answer 2

$\displaystyle a).2+4+6+...+98+100=2(1+2+3+...+49+50)= \\ \\ =2 \times \frac{50(50+1)}{2} =2 \times \frac{50 \times 51}{2} =\not 2 \times \frac{2550}{ \not 2} =2550 \\ \\ b).3+6+9+...+180=3(1+2+3+...+60)=3 \times \frac{60(60+1)}{2} = \\ \\ =3 \times \frac{60 \times 61}{2} =3 \times \frac{3660}{2} =3 \times 1830=5490 \\ \\ c).4+8+12+...360=4(1+2+3+...+90)=4 \times \frac{90(90+1)}{2} = \\ \\ =4 \times \frac{90 \times 91}{2} =4 \times \frac{8190}{2} =4 \times 4095=16380$

$\displaystyle d).5+10+15+...+625=5(1+2+3+...+125)= \\ \\ =5 \times \frac{125(125+1)}{2} =5 \times \frac{125 \times 126}{2} =5 \times \frac{15750}{2} = \\ \\ =5 \times 7875=39375$

$\displaystyle e).1+3+5+...+201= \\ \\ =1+2+3+4+5+...+201-(2+4+6+...+200)= \\ \\ = \frac{201(201+1)}{2} -2(1+2+3+...+100)= \\ \\ = \frac{201 \times 202}{2} -2 \times \frac{100(100+1)}{2} = \frac{40602}{2} -2 \times \frac{100 \times 101}{2} = \\ \\ =20301- \not 2 \times \frac{10100}{\not 2} =20301-10100=10201$

$\displaystyle f).1+4+7+...+301= \\ \\ =1+(1+3)+(1+2 \times 3)+(1+3 \times 4)+...+(1+3 \times 100)= \\ \\ =1 \times 101+3+3 \times 2+3 \times 3+...+3 \times 100= \\ \\ =101+3+6+9+...+300=101+3(1+2+3+...+100)= \\ \\ =101+3 \times \frac{100(100+1)}{2} =101+3 \times \frac{100 \times 101}{2} = \\ \\ =101+3 \times \frac{10100}{2} =101+3 \times 5050=101+15150=15251$

$\displaystyle g).2+7+12+...+177 \\ \\ 177=2+(n-1) \times 5 \\ \\ 177=2+5n-5 \\ \\ -5n=2-5-177 \\ \\ -5n=-180 \\ \\ n= \frac{-180}{-5} \\ \\ n=36$

$\displaystyle S_3_6= \frac{4+35 \times 5}{2} \times 36 \\ \\ S_3_6= \frac{4+175}{\not2} \times \not36 \\ \\ S_3_6= 179 \times 18 \\ \\ S_3_6=3222$

$\displaystyle h).3+7+11+...+199 \\ \\ 199=3+(n-1) \times 4 \\ \\ 199=3+4n-4 \\ \\ 4n=199-3+4 \\ \\ 4n=200 \\ \\ n= \frac{200}{4} \\ \\ n=50$

$\displaystyle S_5_0= \frac{6+49 \times 4}{2} \times 50 \\ \\ S_5_0= \frac{6+196}{2} \times 50 \\ \\ S_5_0= \frac{202}{2} \times 50 \\ \\ S_5_0=101 \times 50 \\ \\ S_5_0=5050$

$\displaystyle i)1+6+11+...+2006+2011 \\ \\ 2011=1+(n-1) \times 5 \\ \\ 2011=1+5n-5 \\ \\ 5n=2011-1+5 \\ \\ 5n=2015 \\ \\ n= \frac{2015}{5} \\ \\ n=403$

$\displaystyle S_{403}= \frac{2+402 \times 5}{2} \times 403 \\ \\ S_{403}= \frac{2+2010}{2} \times 403 \\ \\ S_{403}= \frac{2012}{2} \times 403 \\ \\ S_{403}=1006 \times 403 \\ \\ S_{403}=405418$

$\displaystyle j)1+4+7+...+2011+2014 \\ \\ 2014=1+(n-1) \times 3 \\ \\ 2014=1+3n-3 \\ \\ 3n=2014-1+3 \\ \\ 3n=2016 \\ \\ n= \frac{2016}{3} \\ \\ n=672$

$\displaystyle S_{672}= \frac{2+671 \times 3}{2} \times 672 \\ \\ S_{672}= \frac{2+2013}{2} \times 672 \\ \\ S_{672}= 2015 \times 336 \\ \\ S_{672}= 677040$

$\displaystyle k).4+9+14+...+2009+2014 \\ \\ 2014=4+(n-1) \times 5 \\ \\ 2014=4+5n-5 \\ \\ 5n=2014-4+5 \\ \\ 5n=2015 \\ \\ n= \frac{2015}{5} \\ \\ n=403$

$\displaystyle S_{403}= \frac{8+402 \times 5}{2} \times 403 \\ \\ S_{403}= \frac{8+2010}{2} \times 403 \\ \\ S_{403}= \frac{2018}{2} \times 403 \\ \\ S_{403}=1009 \times 403 \\ \\ S_{403}=406672$