Question

Which term of the AP $72,68,64\ldots..$ is $0$ ?

Answer 1

In this question, we need to find which term of the AP series is $0$ . Given the AP series is $72,68,64,60\ldots$ Here AP stands for arithmetic progression. It is nothing but a sequence where the difference between the consecutive terms are the same. From the given series, we can find the first term $(a)$ and the common difference $(d)$ . Then , by using the formula of arithmetic progression, we can easily find $n$. Here $a_{n}$ is the term for which we need to find the value of $n$, which is given as $0$ .

Formula used:
The Formula used to find the $n^{th}$ terms in arithmetic progression is,
$a_{n} = \ a\ + \ \left( n\ \ 1 \right) \times \ d$
Where $a$ is the first term, $d$ is the common difference, $n$ is the number of terms and $a_{n}$ is the $n^{th}$ term.

Complete step by step answer:
Given $72,68,64,60\ldots$. The general form of the arithmetic sequence can be written as $\\{ a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots\ \\}$
Thus we can tell that $a$ is $72$ and $d$ is $(68 – 72)$ which is $- 4$ . The formula used to find the $n^{th}$ terms in arithmetic progression is
$a_{n} = \ a\ + \ \left( n\ \ 1 \right) \times \ d$
Given that $a_{n}$ is $0$. Now on substituting the known values, we get,
$\Rightarrow \ 72 + (n – 1)\ \times ( - 4)\ = 0$
On simplifying, we get,
$\Rightarrow \ 72 + ( - 4n + 4)\ = 0$

On removing the parentheses, we get,
$\Rightarrow \ 72 – 4n + 4 = 0$
On simplifying, we get,
$- 4n + 76 = 0$
On subtracting both sides by $76$ , we get
$\Rightarrow \ - 4n = - 76$
Now on dividing both sides by $- 4$, we get,
$\Rightarrow \ n = \dfrac{- 76}{- 4}$
On simplifying we get,
$\therefore \ n = 19$
Thus we get $19$ term in the AP series is $0$ .

Therefore, $19$ term in the AP series is $0$.

Note: In order to solve these types of questions, we should have a strong grip over arithmetic series. We should be very careful in choosing the correct formula because there is the chance of making mistakes in interchanging the formula of finding term and summation of term. If we try to solve this sum with the formula $S_{n} = \dfrac{n}{2}\left( a + l \right)\$ where $l$ is the last term of the series , then our answer will be totally different and can get confused.