Question

3+6+9....+x= 2460 folosind metoda progresiei aritmetice
va rog sa ma ajutati

Answer 1

3+6+9+...+x=2460

$a_{1}=3 \\ r=3 \\ a_{n}=x \\ S_{n}=2460 \\ \\ a_{n}=a_{1}+(n-1)r \Rightarrow x = 3+3n-3 \Rightarrow x = 3n\Rightarrow n = \frac{x}{3} \\ \\ S_{n}= \frac{(a_{1}+a_{n})\cdot n}{2} \Rightarrow 2460= \frac{(3+x)\cdot \frac{x}{3} }{2} \Rightarrow 2460 = \frac{ x^{2} +3x}{6} \Rightarrow \\ \\ \Rightarrow x^{2} +3x - 2460\cdot 6 = 0 \\ \\ \triangle = 9+4\cdot 6\cdot 2460 = 59040 +9 = 59049 =243^{2}$

$x_{1,2}= \frac{-3\pm 243}{2} \Rightarrow \left \{ {{ x_{1} =120} \atop { x_{2} =-123} (F)} \right. \\ \Rightarrow x = 120$