Question

Calculaţi valoarea lui a din egalitatea:
(a+1) + (a+3) + (a+5) + ……. + (a+101) = 2703​

Answer 1

Răspuns: a = 2

Explicație pas cu pas:

( a + 1 ) + ( a + 3 ) + ( a + 5 ) + ....... + ( a + 101) = 2 703

1 + 3 + 5 + ..... + 101 =

( 101 - 1 ) : 2 + 1 = 51 termeni

51 × a + 51 × ( 1 + 101): 2 = 2 703

51 × a + 51 × 51 = 2 703

51 × a = 2 703 - 2 601

a = 102 : 51

a = 2

Answer 2

 

$\displaystyle\\(a+1)+(a+3)+(a+5)+...+(a+101)=2703\\\text{Calculam numarul de termeni:}\\\\n=\frac{101-1}{2}+1=\frac{100}{2}+1=50+1=\boxed{\bf51~\text{\bf de termeni}}\\\\\underbrace{a+a+a+...+a}_{\bf51~de~termeni}+\underbrace{1+3+5+...+101}_{\bf51~de~termeni}=2703\\\\\text{plicam Gauss.}\\\\51a+\frac{51(101+1)}{2}=2703\\\\51a+\frac{51\times102}{2}=2703\\\\51a+51\times51=2703\\\\51a+51^2=2703\\\\51(a+51)=2703\\\\a+51=\frac{2703}{51}\\\\ a+51=53\\\\a=53-51\\\\\boxed{\bf~a=2}$