A sequence is given by the recursive rule \(t_1 = −5, t_{n+1} = t_n + 3\) for \(n ≥ 1\). Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?

\(t_1 = −5\)
\(t_{n+1} = t_n + 3\)

\(t_1 = −5\)
\(t_2 = t_1 + 3 = −5 + 3 = −2 \)
\(t_3 = t_2 + 3 = −2 + 3 = 1\)
\(t_4 = t_3 + 3 = 1 + 3 = 4\)
\(t_5 = t_4 + 3 = 4 + 3 = 7\)
The first five terms are: −5, −2, 1, 4, 7

Here,
first term, \(a = −5\), and common difference, \(d = 3\)
So, the general term is given by:
\(t_n = a + (n − 1)d\)
⇒ \(t_n = −5 + 3(n − 1)\)
⇒ \(t_n = 3n − 8\)
Set \(t_n = 52\)
⇒ \(3n − 8 = 52\)
⇒ \(3n = 60\)
⇒ \(n = 20\)
Since \(n\) is a positive number, \(52\) is a term of the sequence. It is the \(20^{th}\) term.