Let aₙ be the n-th term of the A.P.
Then for some fixed number d,
[tex]a_{844} = a_{843} + d[/tex]
[tex]a_{844} = (a_{842} + d) + d = a_{842} + 2d[/tex]
[tex]a_{844} = (a_{841} + d) + 2d = a_{841} + 3d[/tex]
and so on.
Notice how on the right side, the subscript of a and the coefficient of d always add up to 844. Follow this pattern all the way down to a₁₉ to get
[tex]a_{844} = a_{19} + 825d[/tex]
We're told that a₁₉ = 844 and a₈₄₄ = 19. Solve for d :
19 = 844 + 825d
825d = -825
d = -1
We can also write aₙ in terms of an arbitrary k-th term, aₖ, using the pattern from before:
[tex]a_n = a_k + (n - k) d[/tex]
Suppose aₖ = 0 for some value of k. Pick any known value of aₙ, replace d = -1, and solve for k :
a₈₄₄ = 0 + (844 - k) • (-1)
19 = k - 844
k = 863
So, a₈₆₃ = 0.
