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

Question

Accepted Answer

> [!TLDR]
> The key idea is to use the Binomial Theorem to expand $(1.1)^{10000}$ and show that even the first few terms already exceed $1000$. The result is that $(1.1)^{10000}$ is **far larger** than $1000$.

## Why This Approach Works

When comparing a huge power like $(1.1)^{10000}$ with a modest number like $1000$, your first instinct might be to think: "1.1 is just a little above 1, so raising it to a high power might still be small." That intuition is wrong — and dangerously so. Exponential growth is deceptive: even a 10% increase per step, repeated 10,000 times, produces a number so large it dwarfs 1000.

The Binomial Theorem lets us write $(1.1)^{10000}$ as $(1 + 0.1)^{10000}$ and expand it into a sum of positive terms. We don't need the whole sum — just enough terms to prove it's already bigger than 1000.

> [!FORMULA]
> $$(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k$$

Here $n = 10000$ and $x = 0.1$.

## Step-by-Step Reasoning

1. **Write the expansion**  
   $$(1.1)^{10000} = (1 + 0.1)^{10000} = \binom{10000}{0} + \binom{10000}{1}(0.1) + \binom{10000}{2}(0.1)^2 + \cdots + \binom{10000}{10000}(0.1)^{10000}$$

Every term is positive, so the sum is greater than any of its parts.

2. **Look at the first two terms**  
   The first term is $1$. The second term is $\binom{10000}{1} 	imes 0.1 = 10000 	imes 0.1 = 1000$.  
   So already:  
   $$(1.1)^{10000} > 1 + 1000 = 1001$$

That's it — we've already beaten 1000.

3. **But wait — is that enough?**  
   The inequality is strict because there are many more positive terms after the second one. Even if we ignore them, the sum of the first two terms alone is $1001$, which is greater than $1000$. …

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

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Step-by-Step Reasoning

Q.Using Binomial Theorem, indicate which number is larger (1.1)10000(1.1)^{10000}(1.1)10000 or 100010001000.

Why This Approach Works

Step-by-Step Reasoning

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

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