Question

please answer this question from the chapter of Real Nos. class 10​

Answer 1

Answer:

4

First we write general term

$a_n = \frac{r}{1 + {r}^{2} + {r}^{4} } \\ \\ = \frac{r}{ {r}^{4} + 2{r}^{2} + 1 - {r}^{2} } \\ \\ = \frac{r}{( {r}^{2} + 1) {}^{2} - {r}^{2} } \\ \\ = \frac{r}{( {r}^{2} + r + 1)( {r}^{2} + r - 1) } \times \frac{2}{2} \\ \\ = \frac{1}{2} . \frac{ ( {r}^{2} + r + 1) - ( {r}^{2} + r - 1) }{( {r}^{2} + r + 1)( {r}^{2} + r - 1)} \\ \\ = \frac{1}{2} ( \frac{1}{ {r}^{2} - r + 1} - \frac{1}{ {r}^{2} + r + 1} )$

Now let's check terms

$a_1 = \frac{1}{2} (1 - \frac{1}{3} )$

$a_2 = \frac{1}{2} ( \frac{1}{3} - \frac{1}{7} )$

$a_3 = \frac{1}{2} ( \frac{1}{7} - \frac{1}{13} )$

As you can see pattern , in our sum all terms will cancel except first and last so

$s = \frac{1}{2} (1 - \frac{1}{ {99}^{2} + 99 + 1 } ) \approx0.5 - 0.0001 = 0.499$