Question 3

We need to find if 97 and 172 can be generated by the given formula.

Step 1 — Check for 97

Let's assume 97 is a term in the sequence. We set the formula equal to 97.

$5n - 3 = 97$

Let's add 3 to both sides.

$5n = 97 + 3$

$5n = 100$

Now we divide both sides by 5.

$n = \frac{100}{5}$

$\boxed{n = 20}$

Since $n = \mathbf{20}$ is a natural number, 97 is a term. It is the 20th term of the sequence.

Step 2 — Check for 172

Let's assume 172 is a term in the sequence. We set the formula equal to 172.

$5n - 3 = 172$

Let's add 3 to both sides.

$5n = 172 + 3$

$5n = 175$

Now we divide both sides by 5.

$n = \frac{175}{5}$

$\boxed{n = 35}$

Since $n = \mathbf{35}$ is a natural number, 172 is a term. It is the 35th term of the sequence.

Answer

(i) 97 is a term of the sequence. (ii) 172 is a term of the sequence.