Approximation Application
Using the first-order binomial approximation for small quantities, the approximate value of (1.002)^{10} computed to a sensible number of decimal places is closest to which figure?
Select the correct option:
Solution
1.020
For small x, the binomial theorem provides the first-order approximation (1 + x)^n ≈ 1 + nx, where higher powers of the tiny quantity x are negligible, a practical JEE Advanced application. Here write 1.002 = 1 + 0.002 with x = 0.002 and n = 10, so (1.002)^{10} = (1 + 0.002)^{10} ≈ 1 + 10(0.002) = 1 + 0.020 = 1.020. The neglected term is C(10, 2)(0.002)^2 = 45 × 0.000004 = 0.00018, which is small enough not to affect the third decimal materially. Hence the approximation is 1.020. Option 1.002 ignores the exponent entirely. Option 1.200 misplaces the decimal by treating x as 0.02. Option 1.010 halves the linear correction incorrectly. The validity rests on x being small so that quadratic and higher terms are negligible. Plausibility check: since the second-order correction 0.00018 is two orders of magnitude smaller than the first-order term 0.020, the linear approximation 1.020 is reliable to three decimal places, confirming the estimate.
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About This Question
- Subject
- mathematics
- Chapter
- binomial theorem and its simple applications
- Topic
- approximation application
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
1.020
For small x, the binomial theorem provides the first-order approximation (1 + x)^n ≈ 1 + nx, where higher powers of the tiny quantity x are negligible, a practical JEE Advanced application. Here write 1.002 = 1 + 0.002 with x = 0.002 and n = 10, so (1.002)^{10} = (1 + 0.002)^{10} ≈ 1 + 10(0.002) = 1 + 0.020 = 1.020. The neglected term is C(10, 2)(0.002)^2 = 45 × 0.000004 = 0.00018, which is small enough not to affect the third decimal materially. Hence the approximation is 1.020. Option 1.002 ignores the exponent entirely. Option 1.200 misplaces the decimal by treating x as 0.02. Option 1.010 halves the linear correction incorrectly. The validity rests on x being small so that quadratic and higher terms are negligible. Plausibility check: since the second-order correction 0.00018 is two orders of magnitude smaller than the first-order term 0.020, the linear approximation 1.020 is reliable to three decimal places, confirming the estimate.
This easy difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of approximation application. It appeared in the 2025 exam.
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