Question

If 19th term of an A.P. is 844 and 844th term is 19 then find the term which is equal to zero?​

Answer 1

Let aₙ be the n-th term of the A.P.

Then for some fixed number d,

$a_{844} = a_{843} + d$

$a_{844} = (a_{842} + d) + d = a_{842} + 2d$

$a_{844} = (a_{841} + d) + 2d = a_{841} + 3d$

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

$a_{844} = a_{19} + 825d$

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:

$a_n = a_k + (n - k) d$

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.