<p>If <em>x</em><sup>4</sup> occurs in the <em>(r + 1)</em><sup>th</sup> term in the expansion of <span>\(\left(x^4 + \dfrac{1}{x^3}\right)^{15}\)</span>, then find the value of <em>r</em>.</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (a + b)^n, the general term is T_{r+1} = C(n,r) × a^{n-r} × b^r. Identify the power of x in the general term and equate it to 4 to find r.
<p><strong>Step 1:</strong> Write the general term in the expansion of $\left(x^4 + \frac{1}{x^3}\right)^{15}$</p><p>$$T_{r+1} = \binom{15}{r}(x^4)^{15-r}\left(\frac{1}{x^3}\right)^r$$</p><p><strong>Step 2:</strong> Simplify the power of x</p><p>$$T_{r+1} = \binom{15}{r} \cdot x^{4(15-r)} \cdot x^{-3r}$$</p><p>$$T_{r+1} = \binom{15}{r} \cdot x^{60-4r-3r}$$</p><p>$$T_{r+1} = \binom{15}{r} \cdot x^{60-7r}$$</p><p><strong>Step 3:</strong> Set the exponent equal to 4</p><p>$$60 - 7r = 4$$</p><p>$$7r = 56$$</p><p>$$r = 8$$</p><p><strong>Verification:</strong> When r = 8, the exponent is 60 - 7(8) = 60 - 56 = 4 ✓</p><p>∴ Answer: <strong>8</strong></p>
Correct Answer: 8