It's about sequences. Help me please !!

Question

if a_n = 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,... find the value of a₅₀₆₀

 

How can I solve?

Answer 1

First, let's notice that the position p of rightmost "1" in the sequence should satisfy the equation:

 

p² +p < 2*n

 

based on the observation that the sum of the elements to the left of that rightmost "1" is

 

s = 0.5(p+1)*p

 

corresponding to the series:

 

1+2+3+4 +... +p

 

Position p must be a whole number calculated as the root of quadratic equation  p² +p -2n =0 rounded down (notice the Int operator):

 

p_round = Int (-p/2 + √(p2/4 +2n))

 

The answer will be found as a difference: n - 0.5p_round(p_{round +1)}

 

In this particular case of n=5060, p_round = 100, thus a_n = 5060 - 0.5*100*101 = 10

 

an = 10 (the answer)