Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0?

Step 1: Identify the first term and common difference

To solve this arithmetic progression (AP) problem, 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. Here, we subtract the first term from the second term, $18 - 21 = -3$.

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 looking for the term $n$ when $a_n = -81$. Substitute the known values of $a$, $d$, and $a_n$ into the formula to form an equation.

Step 3: Solve for n

Now we solve the equation for $n$. First, subtract $21$ from both sides. Then, divide both sides by $-3$. Finally, add $1$ to find the value of $n$. This value represents the position of $-81$ in the arithmetic progression.

Step 4: Check if any term is 0

To determine if $0$ is a term in the AP, we set $a_n = 0$ in the formula for the $n$ -th term, $a_n = a + (n-1)d$. Substitute the first term $a = 21$ and the common difference $d = -3$ into the equation.

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

Solve this equation for $n$. Subtract $21$ from both sides, then divide by $-3$. Add $1$ to the result to find $n$. Since $n$ is a positive integer, $0$ is indeed a term in the AP.

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