Question 5

We will use the given recursive rule to find the terms of the sequence.

Step 1 — Find the first five terms

The first term is given. We use the recursive rule to find the next terms.

$t_1 = -5$

$t_2 = t_1 + 3$ $= -5 + 3$ $= -2$

$t_3 = t_2 + 3$ $= -2 + 3$ $= 1$

$t_4 = t_3 + 3$ $= 1 + 3$ $= 4$

$t_5 = t_4 + 3$ $= 4 + 3$ $= 7$

$\boxed{\text{The first five terms are } -5, -2, 1, 4, 7}$

Step 2 — Check if 52 is a term

This sequence is an arithmetic progression. The first term is $\mathbf{a = -5}$. The common difference is $\mathbf{d = 3}$. The formula for the $n$-th term is $t_n = a + (n-1)d$.

$t_n = -5 + (n-1)3$ $t_n = -5 + 3n - 3$ $t_n = 3n - 8$

Let's see if $\mathbf{52}$ is a term. We set $t_n = 52$.

$52 = 3n - 8$ $52 + 8 = 3n$ $60 = 3n$ $n = \frac{60}{3}$ $n = 20$

Since $\mathbf{n = 20}$ is a natural number, $\mathbf{52}$ is a term. It is the $\mathbf{20^{th}}$ term of the sequence.

$\boxed{52 \text{ is the } 20^{\text{th}} \text{ term of the sequence.}}$

Answer

(i) The first five terms are $\mathbf{-5, -2, 1, 4, 7}$. (ii) Yes, $\mathbf{52}$ is a term of this sequence. (iii) $\mathbf{52}$ is the $\mathbf{20^{th}}$ term.