Question

Va rog ex. 11 punctul a. Urgent!!! Dau coroana!

Answer 1

 

$\displaystyle\bf\\10+15+20+...+2010-9-13-17-...-1609=\\\\=10+15+20+...+2010-(9+13+17+...+1609)\\\\\text{Avem doua siruri.}\\\text{Calculam numarul de termeni la fiecare sir.}\\\\n_1=\frac{2010-10}{5}+1=\frac{2000}{5}+1=400+1=\boxed{\bf401~termeni}\\\\n_2=\frac{1609-9}{4}+1=\frac{1600}{4}+1=400+1=\boxed{\bf401~termeni}$

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$\displaystyle\bf\\ \boxed{\bf~n_1=n_2=401~termeni}\\\\\text{\bf~Egalitatea de termeni ne permite sa grupam toti termenii}\\\text{\bf~in grupe de 2 numere, cate un numar de la fiecare sir in grupa.}\\\\10+15+20+...+2010-9-13-17-...-1609=\\\\=(10-9)+(15-13)+(20-17)+...+(2010-1609)=\\\\=1+2+3+...+401=\frac{401(401+1)}{2}=(Aplocam~Gauss)\\\\=\frac{401(401+1)}{2}=\frac{401\times402}{2}=401\times201=\boxed{\bf80601}$

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Daca am fi calculat fiecare sir separat (cu Gauss) si am fi scazut rezultatele, am fi obtinut acelasi rezultat, dar cu mai multe calcule si la un examen cu un consum de timp mai mare.

 

 

Answer 2

Răspuns:

(10-9)+(15-13)+(20-17)+....+(2010-1609)=80601

Explicație pas cu pas:

1+2+3+....+401=

401x(401+1):2=80601