Find the values of ${(- 1)}^{n} + {(- 1)}^{2 n} + {(- 1)}^{2 n + 1} + {(- 1)}^{4 n + 1}$ , where $n$ is any positive odd integer.

Open in App

Solution

Let $x = {(- 1)}^{n} + {(- 1)}^{2 n} + {(- 1)}^{2 n + 1} + {(- 1)}^{4 n + 1}$
where $n$ is odd.
${(- 1)}^{n} = - 1 {(- 1)}^{2 n} = 1 {(- 1)}^{2 n + 1} = {(- 1)}^{2 n} . (- 1) = - 1 {(- 1)}^{4 n + 1} = {(- 1)}^{4 n} (- 1) = - 1 ∴ x = - 1 + 1 - 1 - 1 = - 2$

Suggest Corrections

Similar questions

1 - 2 n + \frac{2 n (2 n - 1)}{2!} - \frac{2 n (2 n - 1) (2 n - 1)}{3!} + . . . + (- 1)^{n - 1} \frac{2 n (n - 1) . . . (n + 2)}{(n - 1)}

= (- 1)^{n + 1} \frac{(2 n)}{2 (n!)^{2}},

n

k

n

n \sum k = 1 k^{3} . {(\frac{C_{k}}{C_{k - 1}})}^{2}

C_{k} =^{n} C_{k}

\frac{n {(n + 1)}^{2} (n + 2)}{12}

\frac{C_{k}}{C_{k - 1}} = \frac{^{n} C_{k}}{^{n} C_{k - 1}} = \frac{n - k + 1}{k}

Find the sum of $1. n^{2} + 2 (n - 1)^{2} + 3 (n - 2)^{2} + . . . . . . n {.1}^{2}$

n

n \sum r = 1 r^{3} {(\frac{^{n} C_{r}}{^{n} C_{r - 1}})}^{2} = \frac{n {(n + 1)}^{2} (n + 2)}{12}

n

(\sqrt{3} + 1)^{2 n} - (\sqrt{3} - 1)^{2 n}

Join BYJU'S Learning Program