If - p- q - -0 for p- q p- q- W then -r-0- - n- 2r -

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Sum of Coefficients of All Terms

If [ p; q ]...

Question

If $(\begin{matrix} p q \end{matrix})$ =0 for $p < q$ p, $q ϵ W$ then $\infty \sum r = 0$ $(\begin{matrix} n 2 r \end{matrix})$

2^{n}

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2^{n - 1}

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2^{2 n - 1}

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^2 n C_{n}

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Solution

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lim n \to \infty {(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2})}^{n}

p, q > 0

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

\sum \sum 0 \leq r < s \leq n (r + s) C_{r} C_{s}

{(1 + x + x^{2})}^{n} = 2 n \sum r = 0 a_{r} x^{r} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a^{2 n} x^{2 n}

P = a_{0} + a_{3} + a_{6} + . . .

Q = a_{1} + a_{4} + a_{7} + . . .

R = a_{2} + a_{5} + a_{8} + . . .

P, Q, R

\frac{3 m + 2 n}{q + p} - \frac{4 p - 3 n}{q + r} + \frac{2 p + 3 m}{q + m}

P = \int_{0}^{1} x^{n - 1} (1 - x)^{n} d x

l_{2 n - 1} = \int_{0}^{π} (sin x)^{2 n - 1} d x

\frac{l_{2 n - 1}}{P}

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