Binomial Theorem
Sum of Coefficients
Grade 11
Question:
<p>Given that <i>(1+x-2x</i><sup>2</sup><i>)</i><sup>6</sup> <i>= 1 + a</i><sub>1</sub><i>x + a</i><sub>2</sub><i>x</i><sup>2</sup> <i>+ ... + a</i><sub>12</sub><i>x</i><sup>12</sup>. Find <i>a</i><sub>2</sub> <i>+ a</i><sub>4</sub> <i>+ a</i><sub>6</sub> <i>+ ... + a</i><sub>12</sub>.</p>
Step-by-Step Solution
Key Concept: Use substitution of x=1 and x=-1 to isolate even-indexed coefficients by adding the resulting equations.
<p><strong>Step 1:</strong> Put <i>x = 1</i>:</p><p>$$(1+1-2)^6 = 1 + a_1 + a_2 + ... + a_{12}$$</p><p>$$0 = 1 + a_1 + a_2 + ... + a_{12}$$</p><p><strong>Step 2:</strong> Put <i>x = -1</i>:</p><p>$$(1-1-2)^6 = 1 - a_1 + a_2 - a_3 + ... + a_{12}$$</p><p>$$64 = 1 - a_1 + a_2 - a_3 + ... + a_{12}$$</p><p><strong>Step 3:</strong> Add both equations:</p><p>$$64 = 2(1 + a_2 + a_4 + ... + a_{12})$$</p><p><strong>Step 4:</strong> Therefore:</p><p>$$a_2 + a_4 + a_6 + ... + a_{12} = \frac{64-2}{2} = 31$$</p>
Correct Answer: 31