Question

The 9th term of an A.P is 449 and 449th term is 9. The term which is equal to zero is
(a) 501th
(b) 502th
(c) 508th
(d) None of these

Answer 1

Hint: Assume the first term of the A.P. be $a$ and common difference be $d$. Use the formula, ${{T}_{n}}=a+(n-1)d$ to find 9th term and 449th term of the A.P and solve the two equations to find the value of $a$ and $d$. Assume that ${{n}^{th}}$ term of the A.P is zero and then use the general formula of ${{n}^{th}}$ term to determine the value of $n$.

Complete step-by-step answer:

 In this question Arithmetic Progression also known as A.P. will be used. An A.P. is a sequence of numbers such that the difference of any two successive numbers is a constant called common difference of the A.P. An A.P. is generally represented as:
$a,\text{ }a+d,\text{ }a+2d,\text{ }a+3d,\text{ }.........$, where $a$ is the first term and $d$is the common difference of A.P
To find the $n\text{th}$ term we use, ${{T}_{n}}=a+(n-1)d$.
 Now, let us come to the question. We have been given 9th term and 449th term. Therefore,
$\begin{align}
  & {{T}_{9}}=a+(9-1)d \\
 & 449=a+8d.....................(i) \\
\end{align}$
$\begin{align}
  & {{T}_{449}}=a+(449-1)d \\
 & 9=a+448d......................(ii) \\
\end{align}$
Subtracting equation (i) from (ii), we get,
$\begin{align}
  & -440=440d \\
 & \therefore d=-1 \\
\end{align}$
Putting the value $d$ in equation (i), we get,
$a=449+8=457$.
Now, let us assume that the ${{n}^{th}}$ term of the given A.P is zero. Therefore,
$\begin{align}
  & {{T}_{n}}=a+(n-1)d \\
 & 0=457+(n-1)\times (-1) \\
 & 0=457-n+1 \\
 & n=458 \\
\end{align}$
Hence, option (d) is the correct answer.

Note: It is important to note that in the last stage of solution we needed the values of both the unknowns, $a\text{ and }d$, so , we found the values of these unknowns earlier in the solution so that in the last we just have to put the values. Also, we have put ${{T}_{n}}=0$ because we were asked to find the term which was zero.