1. **Problem:** Find the first three terms of $ (1 + 2x)^{12} $ in ascending powers of $ x $. 2. **Formula:** The binomial expansion for $ (a + b)^n $ is given by: $$ \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k $$ where $ \binom{n}{k} = \frac{n!}{k!(n-k)!} $. 3. **Step 1: Calculate the first three terms for $ (1 + 2x)^{12} $** - Term 0: $ \binom{12}{0} 1^{12} (2x)^0 = 1 $ - Term 1: $ \binom{12}{1} 1^{11} (2x)^1 = 12 \times 2x = 24x $ - Term 2: $ \binom{12}{2} 1^{10} (2x)^2 = 66 \times 4x^2 = 264x^2 $ 4. **Answer for part (i):** $$ 1 + 24x + 264x^2 $$ 5. **Problem (ii):** Given the first three terms in the expansion of $ (2 + ax + 4x^2)(1 + 2x)^{12} $ are $ 2 + 51x + bx^2 $, find $ a $ and $ b $. 6. **Step 2: Expand $ (2 + ax + 4x^2)(1 + 2x)^{12} $ up to $ x^2 $ terms** Using the first three terms of $ (1 + 2x)^{12} $ from above: $$ (2 + ax + 4x^2)(1 + 24x + 264x^2) $$ Multiply out terms up to $ x^2 $: - Constant term: $ 2 \times 1 = 2 $ - Coefficient of $ x $: $ 2 \times 24x + a x \times 1 = 48x + a x = (48 + a)x $ - Coefficient of $ x^2 $: $ 2 \times 264x^2 + a x \times 24x + 4x^2 \times 1 = 528x^2 + 24a x^2 + 4x^2 = (528 + 24a + 4) x^2 = (532 + 24a) x^2 $ 7. **Step 3: Equate coefficients to given expansion $ 2 + 51x + bx^2 $** - For $ x $: $ 48 + a = 51 \implies a = 3 $ - For $ x^2 $: $ 532 + 24a = b $ Substitute $ a = 3 $: $$ b = 532 + 24 \times 3 = 532 + 72 = 604 $$ 8. **Final answers:** $$ a = 3, \quad b = 604 $$