AMC 10 Daily Practice - Sequences
Complete problem set with solutions and individual problem pages
Consider sequences of positive integers for which both the following conditions are true:
(a) each term after the second term is the sum of the two preceding terms;
(b) the eighth term is .
How many such sequences are there?
- A.
- B.
- C.
- D.
- E.
Let and be the first and second terms respectively. Then the first eight terms are
Hence we seek solutions to the equation
, (1)
where and are positive integers, since any such solution will generate a sequence of positive integers of the required sort.
Now , which is a multiple of . Therefore is a multiple of , so that a is a multiple of .
Let , where is a positive integer. Then equation (1) becomes
so that
Since and are positive integers there are therefore only two possible values for , namely and . When , we have and . When , we have and . Hence, there are two possible sequences.
