Question:

<p><strong>Step 1:</strong> Expand using binomial theorem:</p><p><span class="math">a_n = \left(1 + \frac{1}{n}\right)^n = \sum_{r=0}^{n} \binom{n}{r}\frac{1}{n^r}</span></p><p><span class="math">= 1 + 1 + \frac{1}{2!}\left(1 - \frac{1}{n}\right) + \frac{1}{3!}\left(1 - \frac{1}{n}\right)\left(1 - \frac{2}{n}\right) + \cdots</span></p><p><strong>Step 2:</strong> Each term is positive, so <span class="math">a_n > 2</span></p><p><strong>Step 3:</strong> As <span class="math">n \to \infty</span>, <span class="math">a_n \to e \approx 2.718</span>, so <span class="math">a_n < 3</span></p><p><strong>Step 4:</strong> Clearly <span class="math">a_n < 4</span></p><p>∴ Options (a), (b), (c) are correct</p>