Ex 8.2, 8 - Find middle terms of (x/3 + 9y)10 - Binomial - Middle term

  1. Chapter 8 Class 11 Binomial Theorem
  2. Serial order wise
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Ex 8.2,8 Find the middle terms in the expansions of ﷐﷐﷐𝑥﷮3﷯+9𝑦﷯﷮10﷯ Number of terms n = 10 Since n is even. There will be one middle term Middle term = ﷐𝑛﷮2﷯ + 1 = ﷐10﷮2﷯ + 1 = 5 + 1 = 6th term. Hence we need to find 6th term. i.e. T6 We know that general term of (a + b)n is Tr+1 = nCr (a)n-r . (b)r 6th term = T6 = T5+1 of expansion ﷐﷐𝑥﷮3﷯+9𝑦﷯10 Hence r = 5 & n = 5 , a = ﷐𝑥﷮3﷯ & b = 9y T5 + 1 = 10C5 ﷐﷐﷐𝑥﷮3﷯﷯﷮10 − 5﷯ . (9y)5 T6 = 10C5 ﷐﷐﷐𝑥﷮3﷯﷯﷮5﷯ . (9)5 . y5 = ﷐10!﷮5!﷐10 −5﷯!﷯ . ﷐﷐﷐𝑥﷮3﷯﷯﷮5﷯. (3)2 × 5 . y5 = ﷐10!﷮5!5!﷯ . ﷐𝑥5﷮35﷯ . 310 . y5 = ﷐10 × 9 × 8 × 7 × 6 × 5!﷮5 × 4 × 3 × 2 × 1 ×5!﷯ . x5 .﷐310﷮35﷯ . y5 = ﷐10 × 9 × 8 × 7 × 6﷮5 × 4 × 3 × 2 × 1 ﷯ . x5 . 35 . y5 = ﷐10 × 9 × 8 × 7 × 6 ×35﷮5 × 4 × 3 × 2 × 1 ﷯ . x5 . y5 = 61236 x5 y5 Hence middle term of ﷐﷐𝑥﷮3﷯+9𝑦﷯10 is 61236 x5 y5

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