If $\Delta$ stands for the operation 'adding first number to twice the second number', then find the value of $(1 \Delta 2) \Delta 3$.

### Concept & Logic The problem introduces a custom mathematical operator $\Delta$. The rule defined is $a \Delta b = a + 2b$. We must apply this rule following the standard order of operations, resolving parentheses first. ### Step-by-Step Solution **Given:** The operation is defined as adding the first number to twice the second number. Expression to evaluate: $(1 \Delta 2) \Delta 3$ **Calculation:** Step 1: Evaluate the inner bracket $(1 \Delta 2)$. Here, $a = 1$ and $b = 2$. $1 \Delta 2 = 1 + 2(2) = 1 + 4 = 5$ Step 2: Substitute this result back into the original expression. The expression becomes $5 \Delta 3$. Here, $a = 5$ and $b = 3$. $5 \Delta 3 = 5 + 2(3) = 5 + 6 = 11$ ### Exam Strategy & Shortcut Custom operator questions are direct substitution problems. The fastest approach is purely mental math. Quickly visualize $1 + 2(2) = 5$, then $5 + 2(3) = 11$. Do not write down the intermediate equations unless absolutely necessary, as it wastes time. ### Common Pitfall The most common mistake is ignoring the parentheses and processing the expression linearly from left to right without fully evaluating the bracket, or confusing the order by adding twice the first number to the second instead. Always resolve brackets first. ### Final Answer Therefore, the correct answer is **11**.