Solución al problema 115

Enunciado del problema 115

Para los 100 primeros números, p(n) vale:

$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$0$
$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$0$
$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$	$0$
$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$	$27$	$0$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$9$	$18$	$27$	$36$	$45$	$54$	$63$	$72$	$81$	$0$

Sumando cada fila

$$\begin{array}{ccc}{S}_0& =& \left(\frac{9+1}{2}\right)9=5\cdot 9\\ S_1& =& \left(\frac{9+1}{2}\right)9=5\cdot 9\\ S_2& =& \left(\frac{18+2}{2}\right)9=10\cdot 9\\ ⋮& ⋮& ⋮\\ S_9& =& \left(\frac{81+9}{2}\right)9=45\cdot 9\end{array}$$

Sumando todas la filas

$$5\cdot 9+5\cdot 9+10\cdot 9+\dots +45\cdot 9=45+5\left(1+2+\dots +9\right)\cdot 9=45+45\left(\frac{1+9}{2}\right)9=$$

$$=45+45\cdot 45=45\cdot 46=2070$$