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Which term of the A.P. $84,80,76.....$ is $0$ ?

Answer
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 Hint: Use the formula for the nth term of an A.P which is given by . Here, put the value of ${{T}_{n}}=0,a=84,d=-4$ to get the value of n. $d$ can be found by subtracting the 2nd term from the first term.

Complete step-by-step Solution:

We have been given an A.P. in the question whose first term is 84 and the nth term is 0.
Before proceeding with the question we must know that A.P is an arithmetic progression where each term can be found by adding a constant number which is called a common difference.
Now we know that the general form of an arithmetic progression is
$a,a+d,a+2d,a+3d.......$
where a is the first term and d is the common difference.
Now, we know that the nth term of the A.P. is given by the formula ${{T}_{n}}=a+\left( n-1 \right)d$, where ${{T}_{n}}$ is the nth term of the A.P.,$a$ is the first term, $n$ is the number of terms and $d$ is the common difference.
The common difference can be found using the below formula.
Common difference = the next term - the preceding term
The A.P. given in the question is $84,80,76....$.we can write the data from A.P. as
First term a = 84,
The nth term ${{T}_{n}}=0$
Common difference $\Rightarrow d=80-84=-4$
We have to find the value of n.
Therefore, we can use the formula ${{T}_{n}}=a+\left( n-1 \right)d$.
Substituting the value of $a,{{T}_{n}},d$ in the above formula, we get:
$\Rightarrow 0=84+\left( n-1 \right)(-4)$
Opening the brackets, we get
$\Rightarrow 0=84-4n+4$
Adding the constant terms and taking the terms with n on one side, we get
$\Rightarrow -88=-4n$
Dividing both sides with -4, we get,
$\Rightarrow n=\dfrac{88}{4}$
$\therefore n=22$
Hence, we have obtained the ${{22}^{nd}}$ term of the A.P as $0$. Therefore, the answer is ${{22}^{nd}}$ term.

Note: Do not get confused when you get the common difference as a negative number. It can be negative or positive. Always remember whenever we are given the value of the term and we are asked to find the position of that term then we shall always use the formula ${{T}_{n}}=a+\left( n-1 \right)d$. Taking the term in the formula as (n+1) instead of (n-1) can lead to the wrong answer.