Question 3

Let's find the $n^{\text{th}}$ term and the recursive rule for this AP.

Step 1 — Find the $n^{\text{th}}$ term

First, we find the first term. Then, we find the common difference. The first term is $a = \textbf{11}$. The common difference $d$ is the difference between consecutive terms.

$d = 8 - 11$

$\boxed{-3}$

Now, we use the formula for the $n^{\text{th}}$ term of an AP. The formula is $t_n = a + (n-1)d$. Let's substitute the values of $a$ and $d$.

$t_n = 11 + (n-1)(-3)$

$= 11 - 3n + 3$

$= 14 - 3n$

$\boxed{t_n = 14 - 3n}$

Step 2 — Write the recursive rule

A recursive rule defines a term using the previous term. We need the first term. We also need the common difference. The first term is $t_1 = \textbf{11}$. Each term is found by adding the common difference to the previous term.

$t_n = t_{n-1} + d$

$t_n = t_{n-1} - 3$

$\boxed{t_1 = 11, \quad t_n = t_{n-1} - 3 \text{ for } n \ge 2}$

Answer

(i) The $n^{\text{th}}$ term is $14 - 3n$. (ii) The recursive rule is $t_1 = 11$, $t_n = t_{n-1} - 3$ for $n \ge 2$.