Answer
(a)
$\text{A recursively defined sequence is a sequence in which each term is}$ $\text{depending on some or all of preceding terms.}$
---
(b)
$$
a_{1}=3,\\ \quad a_{2}=8,\\ \quad a_{3}=19,\\ \quad a_{4}=42
$$
Work Step by Step
(a)
$\text{A recursively defined sequence is a sequence in which each term is}$ $\text{depending on some or all of preceding terms.}$
---
(b)
the first four terms of the sequence recursively defined by
$a_{1}=3$ and $a_{n}=n+2 a_{n-1}$ will be:
$$
a_{1}=3 \quad \quad \text{(Given)}
$$
$$
a_{2}=2+2 a_{2-1} = 2 + (2 \times 3)= 2+ 6 = 8,
$$
$$
a_{3}=3+2 a_{3-1}= 3 + (2 \times 8) = 3 + 16 = 19,
$$
$$
a_{4}= 4+2 a_{4-1} = 4+ (2 \times 19) = 4 + 38= 42
$$
