If [ ] and [ ] are the roots of the equation 375x2 -- 25x -- 2 = 0, then [ _n _r = 1^n ^r …

Question

If $$\alpha $$ and $$\beta $$ are the roots of the equation 375x2
-- 25x -- 2 = 0, then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\alpha ^r}}  + \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\beta ^r}} $$ is equal to :

Accepted Answer

$$\alpha $$ and $$\beta $$ are the two root of 375x2 - 25x - 2 = 0

Both of the roots are lie in (-1, 1) hence sum of given series is finite

$$\mathop {\lim }\limits_{n \to \infty } \left( {{\alpha  \over {1 - \alpha }} + {\beta  \over {1 - \beta }}} \right) = {{\alpha (1 - \beta ) + \beta (1 - \alpha )} \over {(1 - \alpha )(1 - \beta )}}$$

$$ \Rightarrow {{\left( {\alpha  + \beta } \right) - 2\alpha \beta } \over {1 - (\alpha  + \beta ) + \alpha \beta }} = {{25 - 2( - 2)} \over {375 - 25 - 2}} = {{29} \over {348}}$$

Solutions · 1