A sequence is given by the recursive rule $t_1 = -5$, $t_{n+1} = t_{n+3}$ for $n \geq 1$. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?                                                                           [Page No. 179]

Recursive sequence: $t_1 = -5, t_{n+1} = t_n + 3$

First five terms:

$$t_1 = -5$$

$$t_2 = -2$$

$$t_3 = 1$$

$$t_4 = 4$$

$$t_5 = 7$$

First five terms: -5, -2, 1, 4, 7

To check whether 52 is a term:

This is an arithmetic sequence with first term -5 and common difference 3.

$$t_n = -5 + (n - 1) \times 3$$

$$52 = -5 + 3(n - 1)$$

$$57 = 3(n - 1)$$

$$19 = n - 1$$

$$n = 20$$

So, 52 is a term of the sequence, and it is the $20^{th}$ term.