Question

hallar el valor de 5h siendo: H= 1/35+1/63+1/99+.....+1/59*61 ¿que numero sige?

Answer 1

¡Buenas!

$H= \dfrac{1}{35}+ \dfrac{1}{63}+ \dfrac{1}{99}+ \ldots +\dfrac{1}{59\ \cdot\ 61} \\ \\ \\ \mathit{Escribimos\ la\ sucesion\ de\ la\ siguiente\ manera} \\ \\ \\ H= \dfrac{1}{5\ \cdot\ 7}+ \dfrac{1}{7\ \cdot\ 9}+ \dfrac{1}{9\ \cdot\ 11}+ \ldots +\dfrac{1}{59\ \cdot\ 61}$

$H= \dfrac{1}{2} \left( \dfrac{1}{5}- \dfrac{1}{7 \right)} + \dfrac{1}{2} \left( \dfrac{1}{7}- \dfrac{1}{9 \right)} + \dfrac{1}{2} \left( \dfrac{1}{9}- \dfrac{1}{11 \right)} +\ldots+ \dfrac{1}{2} \left( \dfrac{1}{59}- \dfrac{1}{61 \right)} \\ \\ \\ \mathnormal{Factorizamos} \\ \\ \\ H= \dfrac{1}{2} \left( \dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+ \ldots +\dfrac{1}{57}-\dfrac{1}{59}+\dfrac{1}{59}-\dfrac{1}{61} \right)$

$H= \dfrac{1}{2} \left( \dfrac{1}{5}- \dfrac{1}{7} +\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+ \ldots +\dfrac{1}{57}-\dfrac{1}{59}+\dfrac{1}{59}-\dfrac{1}{61} \right) \\ \\ \\ H= \dfrac{1}{2} \left( \dfrac{1}{5} - \dfrac{1}{61} \right) \\ \\ \\ H= \dfrac{28}{305} \\ \\ \\ 5H = 5 \cdot\ \dfrac{28}{305} \\ \\ \\ 5H = \dfrac{28}{61}$

RESPUESTA

$\boxed{ \dfrac{28}{61} }$