Regional 2003 N3 P2

[Lean]

Se considera la sucesión definida por $a_1 = 5$, y $a_{n+1} =a_n + 3n$. Calcular $a_{100}$.

[Gregorio]

Spoiler: mostrar

$a_{100} = a_{99} + 3 \times 99 = a_{98} + 3 \times 98+ 3 \times 99 = ... = a_1 + 3 \times 1 + 3 \times 2 + ... + 3 \times 98 + 3 \times 99 $
$3 \times 99 + 3 \times 98 + 3 \times 97 + ... + 3 \times 1 = 3 \times (1+2+3+...+99)$
$1+2+3+...+99 = 99 \times 100 /2 = 4950$
$a_1 = 5$
$5+3 \times 4950 = 14855 = a_{100}$