Question

Ajutati-ma sa rezolv si eu exercitiile acestea. La scloala nu ni s-a explicat nimic doar am facut un exemplu care nu prea la-m inteles. Explicati-mi si mie.
5+10+15+....+625
1+3+5+......+201
1+4+7+....+301
2+7+12+....+177
3+7+11+....+199
Prin toate 3 modurile. Va rog frumos.

Answer 1

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

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

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

$\displaystyle d).2+7+12+...+177 \\ 177=2+(n-1) \times 5 \\ 177=2+15n-5 \\ 15n=177-2+5 \\ 15n=180 \\ n=180:5 \\ n=36 \\ S_3_6= \frac{4+35 \times 5}{2} \times 36 \\ \\ S_3_6= \frac{4+175}{\not2} \times \not36 \\ \\ S_3_6= 179 \times 18 \\ S_3_6=3222$

$\displaystyle e).3+7+11+...+199 \\199=3+(n-1) \times 4 \\ 199=3+4n-4 \\ 4n=199-3+4 \\ 4n=200 \\ n= 200:4 \\ n=50 \\ S_5_0= \frac{6+49 \times 4}{2} \times 50 \\ \\ S_5_0= \frac{6+196}{2} \times 50 \\ \\ S_5_0= \frac{202}{2} \times 50 \\ \\ S_5_0=101 \times 50 \\ S_5_0=5050$