Are you smarter than A 5th grader- Part 7

Let's take $a_{i}$ a number from the sequence

We have $a_{1}$ = 6 and $a_{2}$ = 14 and $a_{i}$ = $a_{i-1}$ - $a_{i-2}$ ( for i > 1)

Notice that $\sum_{i = 1}^{6}a_{i}$ = 0 and 2010 ≡ 0 [6]

So $\sum_{i = 1}^{2010}a_{i}$ = 0

Explanation :

$a_{3}$ = $a_{2}$ - $a_{1}$

$a_{4}$ = $a_{3}$ - $a_{2}$ = ( $a_{2}$ - $a_{1}$ ) - $a_{2}$ = - $a_{1}$

$a_{5}$ = $a_{4}$ - $a_{3}$ = - $a_{1}$ - ( $a_{2}$ - $a_{1}$ ) = - $a_{2}$

$a_{6}$ = $a_{5}$ - $a_{4}$ = - $a_{2}$ - (- $a_{1}$ ) = - $a_{2}$ + $a_{1}$ = -( $a_{2}$ - $a_{1}$ )= - $a_{3}$

==> $\sum_{i = 1}^{6}a_{i}$ = 0

After every 6 terms we find that the sum is zero.

When we divide 2010 by 6,we find that the remainder is zero.

Hence the sum is 0.

My Python solution.

seq = [6, 14]
while len(seq) < 2010:
    seq.append(seq[-1] - seq[-2])
print(sum(seq))