Ex 8.2, 9 - In expansion of (1 + a)m + n, prove coefficients

Ex 8.2,9 - Chapter 8 Class 11 Binomial Theorem - Part 2


Transcript

Question 9 In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal. We know that General term of expansion (a + b)n is Tr+1 = nCr an–r br For (1 + a)m + n, Putting n = m + n , a = 1, b = a Tr+1 = n + mCr (1)n + m – r (a)r = n + mCr (a)r Finding coefficient of am ar = am r = m Putting r = m in (1) Tm + 1 = n + mCm (a)m = ((𝑛 + π‘š)!)/π‘š!(𝑛 + π‘š βˆ’ π‘š )! (a)m = ((𝑛 + π‘š)!)/(π‘š! (𝑛)!) (a)m Hence, coefficient of am is ((𝒏 + π’Ž)!)/π’Ž!(𝒏)! Finding coefficient of an ar = an r = n Putting r = n in (1) Tn + 1 = n + mCn (a)n = ((𝑛 + π‘š)!)/𝑛!(𝑛 + π‘š βˆ’π‘› )! (a)n = ((𝑛 + π‘š)!)/𝑛!(π‘š)! (a)n Hence, coefficient of an is ((𝒏 + π’Ž)!)/𝒏!(π’Ž)! Hence, Coefficient of am = Coefficient of an Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.