Question

Using Binomial Theorem, indicate which number is larger \((1.1)^{10000}\) or \( 1000.\)

Answer 1

To compare:

(1.1)¹⁰⁰⁰⁰ and 1000

---

Step 1: Rewrite using Binomial Theorem

1.1 = 1 + 0.1

∴ (1.1)¹⁰⁰⁰⁰ = (1 + 0.1)¹⁰⁰⁰⁰

---

Step 2: Expand first few terms

(1 + x)ⁿ = 1 + nx + ⁿC₂x² + …

Using x = 0.1 and n = 10000:

(1.1)¹⁰⁰⁰⁰ = 1 + 10000(0.1) + ¹⁰⁰⁰⁰C₂(0.1)² + …

= 1 + 1000 + ¹⁰⁰⁰⁰C₂(0.01) + …

---

Step 3: Estimate the second term

¹⁰⁰⁰⁰C₂ = (10000 × 9999) / 2 ≈ 5 × 10⁷

So,

¹⁰⁰⁰⁰C₂(0.01) ≈ 5 × 10⁷ × 0.01 = 5 × 10⁵ = 500000

---

Step 4: Compare values

(1.1)¹⁰⁰⁰⁰ > 1 + 1000 + 500000

(1.1)¹⁰⁰⁰⁰ > 501001

Clearly,

501001 > 1000

---

Conclusion:

(1.1)¹⁰⁰⁰⁰ is much larger than 1000.