The sequence an is defined by a1 = 20, a2 = 30, an+1 = 3an - an-1. Find all n for which 5an+1an + 1 is a square.
Solution
Answer: n = 3, 1 + 5 a3a4 = 63001 = 2512.
We show that 1 + 5an+1an = (an + an+1)2 + 501. Induction on n. It is true for n = 1. Suppose it is true for n-1. Then we have an+1 = 3an - an-1. Hence 5an = 2an + an+1 + an-1. Multiplying by (an+1 - an-1) gives 5anan+1 - 5an-1an = 2anan+1 - 2an-1an + an+12 - an-12 . Adding to 1 + 5an-1an = (an-1 + an)2 + 501 gives 1 + 5anan+1 = (an + an+1)2 + 501, which completes the induction.
The difference between m2 and (m+1)2 is 2m+1. So m2 + 501 cannot be a square for m > 250. A trivial induction shows that an+1 > an and a4 = 180, so for n ≥ 4, we have an + an+1 > 360. Thus the only possible solutions are n = 1, 2 or 3. But for n = 1 we have 3001, which is not a square and for n = 2 we have 10501 which is not a square.
© John Scholes
jscholes@kalva.demon.co.uk
16 June 2002