Example 3

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Example, 3 Which is larger (1.01)1000000 or 10,000? (1.01)1000000 = (1 + 0.01)1000000 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)1000000 = 1000000C0 a1000000 b0 + 1000000C1 a999999 b1 +………….. = ﷐1000000﷮0!﷐1000000 − 0﷯!﷯ a1000000 × 1 + ﷐1000000 !﷮1!(1000000 − 1)!﷯ a999999 b + …… = ﷐1000000!﷮1 ×1000000!﷯ a1000000 + ﷐1000000 × 999999 !﷮1 × 999999!﷯ a999999 b + …… = a1000000 + 1000000a999999 b +……………… (a + b)1000000 = a1000000 + 1000000a999999 b +……………… Putting a = 1 & b = 0.01 (1 + 0.01)1000000 = (1)1000000 + 1000000 (1)99999 . (0.01) +……………… = 1 + 1000000 (1) × (0.01) +…………………… = 1 + 10000.00 +………………. = 10001 + …. > 10, 000 Hence , (1.01)1000000 is larger than 10,000