MathDB

Given a positive integer

c

, we construct a sequence of fractions

a_1, a_2, a_3,...

as follows:

\bullet

a_1 =\frac{c}{c+1}

\bullet

to get

a_n

, we take

a_{n-1}

(in its most simplified form, with both the numerator and denominator chosen to be positive) and we add

2

to the numerator and

3

to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator. For example, if we take

c = 20

, then

a_1 =\frac{20}{21}

and

a_2 =\frac{22}{24} = \frac{11}{12}

. Then we find that

a_3 =\frac{13}{15}

(which is already simplified) and

a_4 =\frac{15}{18} =\frac{5}{6}

. (a) Let

c = 10

, hence

a_1 =\frac{10}{11}

. Determine the largest

n

for which a simplification is needed in the construction of

a_n

. (b) Let

c = 99

, hence

a_1 =\frac{99}{100}

. Determine whether a simplification is needed somewhere in the sequence. (c) Find two values of

c

for which in the first step of the construction of

a_5

(before simplification) the numerator and denominator are divisible by

5

Problem Statement