Ex 8.1,10

Transcript

Ex 8.1,10 Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000. (1.1)10000 = (1 + 0.1)10000 We know that (a + b)n = nC0 an b0 + nC1 an – 1 b1 + nC2 an – 2 b2 + …. …. + nCn – 1 a1 bn – 1 + nCn a0 bn Hence (a + b)10000 = 10000C0 (a)10000 b0 + 10000C1 a9999 b1 +………….. = ﷐10000﷮0!﷐10000 − 0﷯!﷯ a 10000 × 1 + ﷐10000 !﷮1!(10000 − 1)!﷯ a9999 (b)1 +…… = ﷐10000!﷮1 ×10000!﷯ a10000 × 1 + ﷐10000 ×9999!﷮1! × 9999!﷯ a9999 b +…………… = a10000 + 10000a9999 b +…………… (a + b)10000 = a10000 + 10000a9999 b1 +…………… Putting a = 1 & b = 0.1 (1 + 0.1)10000 = (1)10000 + 10000 (1)9999 (0.1)1…………… (1.1)10000 = 1 + 10000 (0.1)+…………. = 1 + 1000.0 + …………. = 1001 +………… > 1000 Hence, (1.1)10000 is larger than 1000