NL-NWO 1989

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\, for \,\, n > 2$$ It is further given that $a_4 = 194$ . Calculate $a_5$ .

Given is a square $ABCD$ with $E \in BC$ , arbitrarily. On $CD$ lies the point $F$ is such that $\angle EAF = 45^o$ . Prove that $EF$ is tangent to the circle with center $A$ and radius $AB$ .

Calculate $$\sum_{n=1}^{1989}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$

Given is a regular $n$ -sided pyramid with top $T$ and base $A_1A_2A_3... A_n$ . The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$ , the plane $TA_2A_3$ at $C_2$ , and so on, and finally the plane $TA_nA_1$ at $C_n$ . Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$ 's.

$\forall k\in N \,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$ Prove: $(n(k) + a(k))a(k) = 1$ , for all $k \in N$ , and calculate $\lim_{k \to \infty }a(k)$