Example 2 - Chapter 8 Class 11 Binomial Theorem

Transcript

Example 2 Compute (98)5. (98)5 = (100 – 2)5 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)5 = = ﷐5!﷮0!﷐ 5 − 0﷯!﷯ a5 × 1 + ﷐5!﷮1!﷐ 5 − 1﷯!﷯ a4 b1 + ﷐5!﷮2!﷐ 5 − 2﷯!﷯ a3 b2 + ﷐5!﷮3!﷐ 5 − 3﷯!﷯ a2b3 + ﷐5!﷮4!﷐ 5 − 4﷯!﷯ a b4 + ﷐5!﷮5!﷐ 5 −5﷯!﷯ b5 × 1 = ﷐5!﷮0! × 5!﷯ a5 + ﷐5!﷮1! × 4!﷯ a4 b + ﷐5!﷮2! 3!﷯ a3 b2 + ﷐5!﷮3! 2!﷯ a2b3 + ﷐5!﷮4! 1!﷯ a b4 + ﷐5!﷮5! 0!﷯ b5 = ﷐5!﷮5!﷯ a5 + ﷐5 × 4!﷮4!﷯ a4 b + ﷐5 × 4 × 3!﷮2! 3!﷯ a3 b2 + ﷐5 × 4 × 3!﷮2 × 1 ×3!﷯ a3b2 + ﷐5 × 4 × 3!﷮3! ×1 ×3!﷯ a2b3 + ﷐5 × 4!﷮4!﷯ ab4 + ﷐5!﷮5! ﷯ b5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 Thus, (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 We need to find (100 – 2)5 Putting a = 100 & b = –2 (100 – 2)5 = (100)5 + 5(100)4 (–2) + 10(100)3 (–2)2 + 10 (100)2 (–2)3 + 5 (100) (–2)4 + (–2)5 (98)5 = 10000000000 + 5 (100000000) (–2) + 10 (10002000) (4) + 10(10000) (–8) + 5(100) (16) + (–32) = 10000000000 – 1000000000 + 40000000 – 800000 + 8000 – 32 = 9039207968 So, (98)5 = 9039207968