Find the next three terms in the sequence: 6, 3, 12, 5, 24, 7, 48, 9, ...

The given pattern is $6, 3, 12, 5, 24, 7, 48, 9, \dots$
Observe the terms at odd positions ($1^{st}, 3^{rd}, 5^{th}, 7^{th}$): $6, 12, 24, 48$.
Observe the terms at even positions ($2^{nd}, 4^{th}, 6^{th}, 8^{th}$): $3, 5, 7, 9$.

For the terms at odd positions ($6, 12, 24, 48$): Each term is obtained by multiplying the previous term in this sequence by 2.
$6 \times 2 = 12$
$12 \times 2 = 24$
$24 \times 2 = 48$

For the terms at even positions ($3, 5, 7, 9$): Each term is obtained by adding 2 to the previous term in this sequence.
$3 + 2 = 5$
$5 + 2 = 7$
$7 + 2 = 9$

The last given term is 9, which is the $8^{th}$ term (an even position).
The $9^{th}$ term will follow the rule for odd positions:
$9^{th}$ term $= 48 \times 2 = 96$.

The $10^{th}$ term will follow the rule for even positions:
$10^{th}$ term $= 9 + 2 = 11$.

The $11^{th}$ term will follow the rule for odd positions:
$11^{th}$ term $= 96 \times 2 = 192$.

The next three terms are $96, 11, 192$.

Explanation:
The pattern consists of two interleaved sequences. The terms at odd positions ($1^{st}, 3^{rd}, 5^{th}, \dots$) form a geometric progression where each term is multiplied by 2. The terms at even positions ($2^{nd}, 4^{th}, 6^{th}, \dots$) form an arithmetic progression where 2 is added to each term.

The extended pattern is: $6, 3, 12, 5, 24, 7, 48, 9, \boxed{\text{96, 11, 192}}$.