Which term of the AP : $21, 18, 15, \ldots$ is $-81$? Also, is any term 0? Give reason for your answer.

Step 1: Identify the first term and common difference

To find a specific term in an Arithmetic Progression (AP), we first need to identify the first term ($a$) and the common difference ($d$). The first term is given as 21. The common difference is found by subtracting any term from its succeeding term, so $18 - 21 = -3$.

Step 2: Find the term that is -81

We use the formula for the $n$ -th term of an AP, $a_n = a + (n-1)d$. We are given $a_n = -81$, $a = 21$, and $d = -3$. We substitute these values into the formula to solve for $n$.

Step 3: Solve for n

We simplify the equation by first subtracting 21 from both sides. Then, we divide both sides by -3 to isolate $(n-1)$. Finally, we add 1 to find the value of $n$. This tells us that the 35th term of the AP is -81.

Step 4: Check if any term is 0

Now we need to determine if any term in the AP can be 0. We set $a_n = 0$ in the formula for the $n$ -th term and solve for $n$. If $n$ is a positive integer, then 0 is a term in the AP.

Step 5: Solve for n when $a_n = 0$

Similar to the previous calculation, we simplify the equation by subtracting 21 from both sides and then dividing by -3. The result shows that $n = 8$. Since $n$ is a positive integer, the 8th term of the AP is 0.

Step 6: State the reasons

The term of an AP is always represented by a positive integer. Since we found positive integer values for $n$ in both cases, it confirms that -81 is the 35th term and 0 is the 8th term of the given Arithmetic Progression.

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