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Hint: For an arithmetic progression having it’s first term denoted by ‘a’, common difference denoted by ‘d’, the ${{r}^{th}}$ term of this arithmetic progression is given by the formula ${{a}_{r}}=a+\left( r-1 \right)d$. This formula can be used to solve this question if we find the first term and the common difference of the given A.P. and then substitute ${{a}_{r}}=0$ in the above formula.

Complete step-by-step answer:

Before proceeding with the question, we must know all the formulas that will be required to solve this question. For an arithmetic progression having its first term equal to a and common difference equal to d, the ${{r}^{th}}$ term of this A.P. is given by the formula,

${{a}_{r}}=a+\left( r-1 \right)d...............\left( 1 \right)$

Also, if we are given the first and the second term of the A.P., the common difference d of the A.P. can be found by subtracting the first term of the A.P. by it’s second term…….$\left( 2 \right)$

In the question, we are given an A.P. 21, 18, 15, ……… and we are required to find the number of the term which is equal to 0.

We can see that the first term of the A.P. i.e. $a$ is 21 and the second term of the A.P. is 18. So, using formula $\left( 2 \right)$, the common difference is given by,

d = 18 - 21 = -3

Let us assume ${{r}^{th}}$ term of this A.P. is equal to 0. Substituting ${{a}_{r}}=0,a=21$ and d = -3 in formula $\left( 1 \right)$, we get,

$\begin{align}

& 0=21+\left( r-1 \right)\left( -3 \right) \\

& \Rightarrow 3\left( r-1 \right)=21 \\

& \Rightarrow r-1=7 \\

& \Rightarrow r=8 \\

\end{align}$

Hence, the ${{8}^{th}}$ term of the A.P. is equal to 0.

Note: There is a possibility that one may commit a mistake while finding the common difference of the given arithmetic progression. There is a possibility that one may find the common difference of this arithmetic as +3 instead of -3. But since the common difference is found by subtracting ${{n}^{th}}$ term from ${{\left( n+1 \right)}^{th}}$ term, the common difference will be equal to -3 and not +3.

Complete step-by-step answer:

Before proceeding with the question, we must know all the formulas that will be required to solve this question. For an arithmetic progression having its first term equal to a and common difference equal to d, the ${{r}^{th}}$ term of this A.P. is given by the formula,

${{a}_{r}}=a+\left( r-1 \right)d...............\left( 1 \right)$

Also, if we are given the first and the second term of the A.P., the common difference d of the A.P. can be found by subtracting the first term of the A.P. by it’s second term…….$\left( 2 \right)$

In the question, we are given an A.P. 21, 18, 15, ……… and we are required to find the number of the term which is equal to 0.

We can see that the first term of the A.P. i.e. $a$ is 21 and the second term of the A.P. is 18. So, using formula $\left( 2 \right)$, the common difference is given by,

d = 18 - 21 = -3

Let us assume ${{r}^{th}}$ term of this A.P. is equal to 0. Substituting ${{a}_{r}}=0,a=21$ and d = -3 in formula $\left( 1 \right)$, we get,

$\begin{align}

& 0=21+\left( r-1 \right)\left( -3 \right) \\

& \Rightarrow 3\left( r-1 \right)=21 \\

& \Rightarrow r-1=7 \\

& \Rightarrow r=8 \\

\end{align}$

Hence, the ${{8}^{th}}$ term of the A.P. is equal to 0.

Note: There is a possibility that one may commit a mistake while finding the common difference of the given arithmetic progression. There is a possibility that one may find the common difference of this arithmetic as +3 instead of -3. But since the common difference is found by subtracting ${{n}^{th}}$ term from ${{\left( n+1 \right)}^{th}}$ term, the common difference will be equal to -3 and not +3.

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