Question 2

We will use the formula for the $n$-th term of an Arithmetic Progression.

Step 1 — Find the general term

Let's identify the first term and common difference. The given AP is 21, 18, 15, .... The first term, $a$, is 21. The common difference, $d$, is found by subtracting a term from its next term.

$d = 18 - 21$

$\boxed{d = -3}$

Now, we write the formula for the $n$-th term, $t_n$. The formula is $t_n = a + (n - 1)d$. Let's substitute the values of $a$ and $d$.

$t_n = 21 + (n - 1)(-3)$

$t_n = 21 - 3n + 3$

$\boxed{t_n = 24 - 3n}$

Step 2 — Check for -81

We want to find if -81 is a term. Let's set $t_n$ equal to -81.

$24 - 3n = -81$

$-3n = -81 - 24$

$-3n = -105$

$n = \frac{-105}{-3}$

$\boxed{n = 35}$

Since n = 35 is a natural number, -81 is a term. It is the 35th term of the AP.

Step 3 — Check for 0

Now, let's see if 0 is a term. We set $t_n$ equal to 0.

$24 - 3n = 0$

$24 = 3n$

$n = \frac{24}{3}$

$\boxed{n = 8}$

Since n = 8 is a natural number, 0 is a term. It is the 8th term of the AP.

Answer

(i) The 35th term of the AP is -81. (ii) Yes, 0 is a term of this AP. (iii) 0 is the 8th term of the AP.