Răspuns:
x=29
Explicație pas cu pas:
x=?
1+3+5+...+x=225
pasi de rezolvare:
- construim suma lui Gauss:
- adaugam termenii lipsa, apoi ii scadem:
1+2+3+4+5+6+...+(x-1)+x-2-4-6-...(x-1)=225
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De unde x-1?
Observam ca suma initala e formata doar din termeni impari => x va fi termen impar
Mie imi lipsesc numerele pare... pe care trebuie sa le adaug si sa le scad ulterior....ok, dar... pana unde adaug? Pana la cel mai mare nr par... si ala va fi x-1
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1+2+3+4+5+6+...+(x-1)+x-2-4-6-...(x-1)=225
2. aplic formula sumei lui Gauss si dau un doi factor comun din numerele pare scazute (adaug si scad dupa pentru a nu mi se schimba valoarea sumei S+a-a=S)
FORMULA:
[tex]S=\frac{x(x+1)}{2}[/tex], unde S= 1+2+3+4+....+(x-1)+x
1+2+3+4+5+6+...+(x-1)+x-2-4-6-...(x-1)=225
[tex]\bold{\frac{x(x+1)}{2}} -2(1+2+3+....+\frac{x-1}{2} )=225[/tex]
3. Aplicam iar suma lui Gaus:
[tex]\frac{x(x+1)}{2} - 2\frac{\frac{(x-1)}{2}*\frac{(x-1)+2}{2}}{2}=225\\\\ \frac{x(x+1)}{2}-\frac{x-1}{2}*\frac{x+1}{2}=225\\\\\frac{x(x+1)}{2}-\frac{(x-1)(x+1)}{2*2}=225\\\\\frac{x(x+1)}{2}-\frac{x^{2} -1}{4} =225\\\\\frac{x^{2} + x}{2}-\frac{x^{2} -1}{4} =225\\\\\frac{2x^{2} + 2x}{4}-\frac{x^{2} -1}{4} =225\\\\\frac{2x^{2} + 2x-x^{2} +1}{4} =225\\\\\frac{x^{2} + 2x +1}{4}=225\\\\x^{2} + 2x +1 = 225*4\\\\(x+1)^{2} =900\\x - nr \ natural \\= > x+1=30= > \boxed{\boxed{x=29}}[/tex]
