Question 10 of 20intermediate🔧 ApplyNumerical4 marks

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

Correct Answer

(1.1)^{10000} > 1000. Therefore, (1.1)^{10000} is larger than 1000.

Exercise: EXERCISE 7.1 | Q: 10 | (Page 8)
For More Understanding

Explanation

This question follows the same pattern as Example 3 in the textbook. We need to express 1.1 as (1 + 0.1) and apply the Binomial Theorem. The key insight is that when expanding (1 + 0.1)^{10000}, all terms are positive, so we can compare with 1000 by looking at just the first few terms of the expansion.

Solution Steps

  1. Step 1: Express (1.1)^{10000} in binomial form: (1.1)^{10000} = (1 + 0.1)^{10000}

  2. Step 2: Apply Binomial Theorem: (1 + 0.1)^{10000} = ^{10000}C_0(1)^{10000} + ^{10000}C_1(1)^{9999}(0.1) + other positive terms

  3. Step 3: Simplify the first two terms: = 1 + 10000×0.110000 \times 0.1 + other positive terms

  4. Step 4: Calculate: = 1 + 1000 + other positive terms

  5. Step 5: Since all terms in the expansion are positive (as 1 > 0 and 0.1 > 0), we have (1.1)^{10000} = 1 + 1000 + other positive terms > 1000

  6. Step 6: Conclusion: (1.1)^{10000} > 1000

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