Question

The 9th term of an A.P. is 499th term is 9. Which term of this A.P.is zero?

Answer 1

$\huge\bold\green{Answer:-}$

508

$\textbf{Step by step explanation}$

Note:- Error in the question

Correct question is :-

$\textbf{The 9th term of an A.P is 449 and 449th term is 9. Which term of this A.P is zero?}$

Solution :-

Given

a9 = 499

A449 = 9

Let first term be a

and common difference be d

A / q

a9 = 499

=> a + 8d = 499 --------- ( 1 )

And

A499 = 9

=> a + 498d = 9 -----------( 2 )

On solving ( 1 ) and ( 2 ), we get

a + 8d = 499

a + 498d = 9
--------------------

=> - 490d = 490

=> d = - 1

From ( 1 )

a + 8d = 499

=> a + 8 ( - 1 ) = 499

=> a - 8 = 499

=> a = 499 + 8

=> a = 507

Let nth term be 0

We know that nth term of an AP is given by

an = a + ( n - 1 ) * d

$\underline{Put \: the\: value\: of \:a\: and\: d}$

=> 0 = 507 + ( n - 1 ) * ( - 1 )

=> 0 = 507 - n + 1

=> 0 = 508 - n

=> 0 - 508 = - n

=> - 508 = - n

=> n = 508.

$\textbf{Hence 508th term of this AP is zero}$

Hope it helps:-)

Answer 2

$\huge\bold\pink{Let a is the first term and d is the common difference of the A.P}$

$\huge\bold\brown{Given 9th term of AP = 499}$

a = 8d = 499 ==> eqn 1

499th term of AP = 9

a + 498d = 9 ==> eqn 2

Now differentiate eqn 1 and 2

$\textbf{a + 8d - ( a + 498d ) = 499 - 9}$

$\textbf{a + 8d - a - 498d = 490}$

$\texbf{-490d = -490 / 490}$

$\texbf{d = -1}$

Now put "d" value in eqn 1

$\texbf{a - 8 = 499}$

$\texbf{a = 499 + 8}$

$\texbf{a = 507}$

let nth term is equal to zero

$\huge\bold\red{a + ( n - 1 ) d = 0}$

$\huge\bold\red{507 + ( n - 1 ) -1 = 0}$

$\huge\bold\red{507 - ( n - 1 ) = 0}$

$\huge\bold\red{507 - n + 1 = 0}$

$\huge\bold\red{508 - n = 0}$

$\huge\bold\black{n = 508}$

So ,

508th term of AP is equal to Zero .