If x square +x+1=0 then find the value of summasion of x^r +1÷x^r

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lim x \to 0 \frac{100 \sum r = 1 [x^{r}]}{1 + | x |} =

Q. If the coefficient of

x^{r - 1}

x^{r}

x^{r + 1}

in the expansion of

{1 + x}^{n}

are in AP.then

n^{2} - (4 r + 1) n + 4 r^{2} - 2 = 0

If the coefficients of $x^{r - 1}, x^{r}$ and $x^{r + 1}$ in the binomial expansion of $(1 + x)^{n}$ are in AP, prove that
$n^{2} - n (4 r + 1) + 4 r^{2} - 2 = 0$

if $x_{r}$ =cos $(\frac{π}{2^{r}})$ +isin $(\frac{π}{2^{r}})$ , then $x_{r}$ , $x_{r}$ , ...... $\infty$ is

Q. If

C_{r}

is coefficient of

x^{r}

in expansion of

{(1 + x)}^{n}, n ϵ N, n \geq 4

then

n \sum r = 0 {(- 1)}^{r} (^{n} C_{r}) \frac{(1 + r ln 10)}{{(1 + ln 10^{n})}^{r}}

is divisible (remainder 0) by
(1+x)

^{n}

, n

ϵ N

, n

\geq 4

c_{r}, x^{r}

n \sum r = 0 {(- 1)}^{r} (^{n} C_{r}) \frac{(1 + r ln 10)}{{(1 + ln 10^{n})}^{r}} = 0

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