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

By establishing two equations based on the given 19th and 844th term of the A.P. and solving for the first term 'a' and the common difference 'd', it was determined that the 863rd term of the A.P. is zero.

Step-by-step explanation:

In an arithmetic progression (A.P.), each term after the first is obtained by adding a constant, known as the common difference, to the previous term. The nth term (Tn) of an A.P. is given by the formula Tn = a + (n-1)d, where 'a' is the first term and 'd' is the common difference.

Given that the 19th term is 844, we have T19 = a + (19-1)d = a + 18d = 844. Similarly, for the 844th term, T844 = a + (844-1)d = a + 843d = 19.

To find the term which is equal to zero, we need to solve for 'n' when Tn = 0, using the same formula for the nth term of an A.P. By solving these equations simultaneously, we can find the values of 'a' and 'd', and then use them to find the term number which equals zero.

Let's set up the equations:

1. a + 18d = 844 (1)
2. a + 843d = 19 (2)

Subtracting equation (1) from equation (2), we can find the common difference 'd':

843d - 18d = 19 - 844

825d = -825

d = -1

Now, using 'd' in equation (1):

a - 18 = 844

a = 862

Finally, to find the term that equals zero:

862 + (n-1)(-1) = 0

862 - n + 1 = 0

n = 863

So, the 863rd term of the A.P. is zero.

Answer 2

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.