1. The problem states that for any two numbers $a$ and $b$, the operation $\Delta$ is defined as $a \Delta b = (a \times b) + a$. 2. We need to find the value of $3 \Delta (4 \Delta 1)$. 3. First, calculate the inner operation $4 \Delta 1$: $$4 \Delta 1 = (4 \times 1) + 4 = 4 + 4 = 8$$ 4. Now substitute this result back into the outer operation: $$3 \Delta 8 = (3 \times 8) + 3 = 24 + 3 = 27$$ 5. Therefore, the value of $3 \Delta (4 \Delta 1)$ is $27$. Final answer: 27