Question

Rezolvati in R ecutia x+4x+7x+10x+...+151x=11628
Va rog frumos, urgeeeentt!!!!
Dau 50 puncte!

Answer 1

x(1+4+7+10+ ... +151)=11628

Obs:
1=3·0+1
4=3·1+1
7=3·2+1
10=3·3+1
.......................
151=3·50+1


x(1+4+7+10+...+151)=11628
x(3·0+1 + 3·1+1 + 3·2+1 + 3·3+1 +.... + 3·50+1)=11628
x[3(0+1+2+3+...+50)+1·51]=11628
x[3· (50·51:2) +51]=11628
x(3·25·51 +51)=11628
x(75·51+51)=11628
( x[ 51(75+1)]=11628 )
x·76·51=11628 |:76
x·51=153 |:51
x=3


R: x=3

Answer 2

Met 1 prin progresii aritmetice:
------------------------------------------
x(1+4+7+10+...151)=11628
Progresie aritmetica de ratie 3 si termeni=151-1  /  3     +    1 = 150/3 + 1 = 50+1=51.
S= 51(1+151)/2 = 51*152/2=51*76
51*76x=11628
3*17* 2^2 * 19 * x = 2^2 * 3^2 * 17 * 19 l : 2^2 * 3 * 17 * 19
x=3
--------------------
Met 2 prin suma Gauss:
x(3*0+1 + 3*1+1  + 3*2 +1  + 3*3 +1 +...+ 3*50 +1) = 11628
x[3(0+1+2+3+...+50) +1*51] = 11628
x[3*50(1+50)/2 + 3*17] = 11628
x(3*25*51 + 51) = 11628
51x(75+1) = 11628 l : 51
76x = 228
x= 228:76 = 3