# PROBLEM 1 

Consider the sequence of fractions: $\frac{1}{1 \cdot 2}, \frac{2}{2 \cdot 4}, \frac{3}{4 \cdot 7}, \frac{4}{7 \cdot 11}, \frac{5}{11 \cdot 16}, \ldots .$.

a) Complete the sequence with the next two fractions;

b) Calculate the sum of the first 10 fractions in the sequence;

c) What is the 100th fraction in the sequence?
## Solution:

a) $\frac{6}{16 \cdot 22}, \frac{7}{22 \cdot 29}$ 2p
b) $S=\frac{1}{1 \cdot 2}+\frac{2}{2 \cdot 4}+\ldots+\frac{10}{46 \cdot 56}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}-\ldots+\frac{1}{46}-\frac{1}{56}=1-\frac{1}{56}=\frac{55}{56} \quad$ 3p
c) $T_{100}=\frac{100}{a \cdot b}$ with $a=b-100$

$1 p$

$b=1+1+2+3+4+\ldots .+100=1+\frac{100 \cdot 101}{2}=5051$

$1 \mathrm{p}$