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18.03.2013

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, .

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.

1. .

k = 1, = 2cosπ/4. .

2. .

k = n (n ∈ N)

. , k = n + 1, ..

( )

:

= 2cosπ/2n+2.

:

2 + 2cosπ/2n+1 = 4cos2 π/2n+2.

2cos2x = cos2x + 1.

4cos2 π/2n+2 = 2(cosπ/2n+1 + 1) = 2cosπ/2n+1 + 2. .

, nN.


nxn+1 - (n+1)xn + 1 = 0.

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:

nxn+1 - (n+1)xn + 1 = nxn(x - 1) - (xn - 1) = (x - 1)(nxn - xn-1 - xn-2 - ... - 1) = 0.

, x = 1 - . , .

, x > 1, xn > xi, i < n,

nxn - xn-1 - xn-2 - ... - 1 > nxn - nxn-1 > 0.

0 < x < 1, xn < xi, i < n,

nxn - xn-1 - xn-2 - ... - 1 < nxn - nxn-1 < 0.

.

: x = 1.


x, y, z x3 + y3 = z3. , 3.

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. x, y, z 3. x = 3a 1, y = 3b 1, z = 3c 1. :

(3a 1)3 + (3b 1)3 = (3c 1)3;

27a3 27a2 + 9a 1 + 27b3 27b2 + 9b 1 = 27c3 27c2 + 9c 1;

9(3a3 3a2 + a + 3b3 3b2 + b - 3c3 -+ 3a2 - c) = -+ 1 1 -+ 1.

9. . . 3.

.


1/12 + 1/23 + ... + 1/n(n + 1),   nN.

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:

1/k - 1/(k + 1) = 1/k(k + 1).

:

1/12 + 1/23 + ... + 1/n(n + 1) = 1/1 - 1/2 + 1/2 - 1/3 + ... + 1/n - 1/(n + 1) = 1 - 1/(n + 1).

: 1 - 1/(n + 1).


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