Scrumptious Sequence

Number Theory Level 3

A sequence is as follows:

1,0,1,0,1,0,3,5,0,9...

such that every term (except the first six terms) is equal to the last digit of the sum of the last six terms.

Is it true that the sequence contains six consecutive terms equal to 0,1,0,1,0,1?

The sequence cannot be continued Yes Cannot be Determined No

1 solution

Rajdeep Ghosh
Jul 16, 2017

This is one of the sweetest solutions I've ever seen. It is due to one of the INMO trainers who have taught me.

Define $n_i$ as the $i^{th}$ term of the sequence.

Also, define $X_i= n_i+2n_{i+1}+3n_{i+2}+4n_{i+3}+5n_{i+4}+6n_{i+5}$

Now, $X_i\equiv n_i+2n_{i+1}+3n_{i+2}+4n_{i+3}+5n_{i+4}+6n_{i+5} \mod 5$

or, $X_i\equiv n_i+2n_{i+1}+3n_{i+2}+4n_{i+3}+5n_{i+4}+6(n_{i-1}+n_{i}+n_{i+1}+n_{i+2}+n_{i+3}+n_{i+4})\mod 5$

or, $X_i\equiv n _{i-1}+2n_{i}+3n_{i+1}+4n_{i+2}+5n_{i+3}+6n_{i+4}\mod 5$

or, $X_i\equiv X_{i-1}\mod 5$

So, all $X_i$ s are congruent to each other $\mod5$ .

Initially, $X_i\equiv 4\mod 5$ .

If the terms 0,1,0,1,0,1 exist consecutively then $X_i\equiv 12\equiv 7\mod 5$ .

So, the six terms 0,1,0,1,0,1 cannot occur consecutively.

(P.S-I still don't know how he came up with the idea of suddenly defining $X_i$ )