Question

Which term of AP: 22, 19, 16 ..., is its first negative term?

Answer 1

Hint: The \[{{n}^{th}}\] term of an arithmetic progression can be written as follows
\[=a+(n-1)d\]
(Where a is the first term of an arithmetic progression and d is the common difference of the arithmetic progression)
The first term of an arithmetic progression which is negative has to be less than 0 and so for finding it, we can simply form an inequality to get to the solution.

Complete step-by-step solution -

As mentioned in the question, we have to find the first negative term of the given arithmetic progression.
Now, as we know the formula for finding the \[{{n}^{th}}\] term of an arithmetic progression that is mentioned in the hint, we can write the general formula for this given arithmetic progression as follows
\[\begin{align}
  & =a+(n-1)d \\
 & =22+(n-1)(-3) \\
 & =22-3n+3 \\
 & =-3n+25 \\
\end{align}\]
(As in the arithmetic progression that is given, we have the first term as 22 that means a=22 and by subtracting the second and the first term, we get the common difference as -3, that is d=-3 )
Now, for finding the first negative term we can do the following
\[\begin{align}
  & -3n+25<0 \\
 & 25<3n \\
 & n>\dfrac{25}{3} \\
 & n>8.334 \\
\end{align}\]
Hence, the 9th term of the given series will be the first negative term of this arithmetic progression.

Note: The students can make an error if they don’t know how to write the general term of an arithmetic progression which can be written as follows
\[\begin{align}
  & general\ term=a+(n-1)d \\
 & general\ term=(a+d)+nd \\
\end{align}\]