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Number theory Difficulty 4.1 AIME Prove it Canada

In an Ing sequence, the first term is a positive integer
and each term after the first is determined in the following way:

if a term, $x$ , is odd, the
next term is $x+3$ , and
if a term, $x$ , is even, the
next term is $x+4$ .

For example, if the first term in an Ing sequence is 13, then the
second term is 16, and the third term is 20.

If the first term in an Ing sequence is
7, what is the fifth term in the sequence?
If the fifth term in an Ing sequence is
62, what are the two possibilities for the first term?
If the first term in an Ing sequence is
49, determine the terms appearing in the sequence whose values are
greater than 318 and less than 330.
The number 18 appears somewhere in an Ing
sequence after the first term. If the first term is the positive integer
$n$ , determine all possible values of
$n$ .

Solution

If the first term is 7, then the second term is $7+3=10$ (since 7 is odd, we add 3).

If the second term is 10, then the third term is $10+4=14$ (since 10 is even, we add
4).

Similarly, the fourth term is $14+4=18$ , and the fifth term is $18+4=22$ .

If the first term in an Ing sequence is 7, then the fifth term in the
sequence is 22.
Suppose that a term, $x$ , is
odd. The next term, $x+3$ , is even
since an odd integer plus an odd integer is even.

Suppose that a term, $x$ , is even.
The next term, $x+4$ , is even since
an even integer plus an even integer is even.

This means that in an Ing sequence, each term after the first is an even
integer.

Thus, if the fifth term is 62, then the fourth term cannot equal $62-3=59$ (since 59 is odd), and so it must
equal $62-4=58$ .

Similarly, the third term is $58-4=54$ , and the second term is $54-4=50$ .

If the first term is an even integer, then the first term is $50-4=46$ , and if it is an odd integer,
then the first term is $50-3=47$ .

If the fifth term in an Ing sequence is 62, then the first term is 46
(the terms are 46, 50, 54, 58, 62) or the first term is 47 (the terms
are 47, 50, 54, 58, 62).
If the first term is 49, then the second term is $49+3=52$ , and so each term after the
second is 4 more than the previous term.

This means that for every positive integer $k$ , there is a term of the form $52+4k$ in the sequence.

Since $52+4k=4(13+k)$ , then each of
the remaining terms in the sequence is a multiple of 4.

The integers that are greater than 318 and less than 330, and that are
equal to a multiple of 4 are $320=4\times80$ , $324=4\times81$ , and $328=4\times82$ .
If 18 appears somewhere in an Ing sequence after the first term,
then the term preceding 18 is either $18-3=15$ , or it is $18-4=14$ .

Each of these is a possible value of $n$ , the first term of the sequence.

As was shown in part (b), each term after the first in an Ing sequence
is even, and so if 15 appears in the sequence, then 15 can only be the
first term of the sequence.

Since 14 is even, then it could be the first term, but it could also be
a term after the first.

If 14 appears in the sequence after the first term, then the preceding
term is either $14-3=11$ , or it is
$14-4=10$ .

Each of these is a possible value of $n$ .

If 11 appears in the sequence, then 11 is the first term (since 11 is
odd).

Since 10 is even, then it could be the first term, but it could also be
a term after the first.

If 10 appears in the sequence after the first term, then the preceding
term is either $10-3=7$ , or it is
$10-4=6$ .

Each of these is a possible value of $n$ .

If 7 appears in the sequence, then 7 is the first term.

Since 6 is even, then it could be the first term, but it could also be a
term after the first.

If 6 appears in the sequence after the first term, then the preceding
term is either $6-3=3$ , or it is
$6-4=2$ .

Each of these is a possible value of $n$ .

If 3 appears in the sequence, then 3 is the first term.

If 2 appears in the sequence, then 2 must also be the first term of the
sequence since both $2-3=-1$ and
$2-4=-2$ are not positive
integers.

Thus, if 18 appears somewhere in an Ing sequence after the first term,
then the possible values of the first term $n$ are 2, 3, 6, 7, 10, 11, 14, and
15.

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