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Example 9 - Coefficients of three consecutive terms (1 + a)n - Examples

  1. Chapter 8 Class 11 Binomial Theorem
  2. Serial order wise
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Example 9 The coefficients of three consecutive terms in the expansion of (1 + a)n are in the ratio 1: 7 : 42. Find n. Let the three consecutive terms be (r – 1)th, rth and (r + 1)th terms. i.e. Tr-1 , Tr & Tr+1 We know that general term of expansion (a + b )n is Tr+1 = nCr an – r br For (1 + a)n , Putting a = 1 , b = a Tr+1 = nCr 1n – r ar Tr+1 = nCr ar ∴ Coefficient of (r + 1)th term = nCr For rth term of (1 + a)n Replacing r with r – 1 in (1) Tr – 1 + 1 = nCr – 1 ar – 1 Tr = nCr – 1 ar – 1 ∴ Coefficient of (r)th term = nCr-1 For (r – 1)th term of (1 + a)n Replacing r with r – 2 in (1) Tr – 2 + 1 = nCr – 2 ar – 2 Tr – 1 = nCr – 2 ar – 2 ∴ Coefficient of (r – 1)th term = nCr – 2 Since the coefficient of (r – 1)th, rth and (r + 1)th terms are in ratio 1 : 7 : 42 ﷐𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ﷐(𝑟−1)﷮𝑡ℎ﷯ 𝑡𝑒𝑟𝑚﷮𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ﷐𝑟﷮𝑡ℎ﷯ 𝑡𝑒𝑟𝑚﷯ = ﷐1﷮7﷯ ﷐﷐𝑛𝐶﷮𝑟 −2﷯﷮﷐𝑛𝐶﷮𝑟 −1﷯﷯ = ﷐1﷮7﷯ ﷐﷐𝑛!﷮﷐𝑟 − 2﷯![𝑛 − ﷐𝑟 − 2)﷯!﷯﷮﷐𝑛!﷮﷐𝑟 − 1﷯!﷐𝑛 − (𝑟 − 1)﷯!﷯﷯ = ﷐1﷮7﷯ ﷐𝑛!﷮﷐𝑟 − 2﷯![𝑛 − ﷐𝑟 − 2)﷯!﷯ × ﷐﷐𝑟 − 1﷯![𝑛 − ﷐𝑟 − 1)﷯!﷮𝑛!﷯ = ﷐1﷮7﷯ ﷐﷐𝑟−1﷯﷐𝑛−﷐𝑟−1﷯﷯!﷮(𝑛−﷐𝑟−2﷯)!﷯ = ﷐1﷮7﷯ ﷐﷐𝑟 − 1﷯﷐𝑛− 𝑟 + 1﷯!﷮(𝑛 − 𝑟 + 2)!﷯ = ﷐1﷮7﷯ ﷐﷐𝑟 − 1﷯﷐𝑛− 𝑟 + 1﷯!﷮(𝑛 − 𝑟 + 2)(𝑛 − 𝑟 +2 −1)!﷯ = ﷐1﷮7﷯ ﷐﷐𝑟 − 1﷯﷐𝑛− 𝑟 + 1﷯!﷮(𝑛 − 𝑟 + 2)(𝑛 − 𝑟 +1)!﷯ = ﷐1﷮7﷯ ﷐﷐𝑟 − 1﷯﷮(𝑛 − 𝑟 + 2) ﷯ = ﷐1﷮7﷯ 7(r – 1) = n – r + 2 n – 8r + 9 = 0 Also ﷐𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ﷐𝑟﷮𝑡ℎ﷯ 𝑡𝑒𝑟𝑚﷮𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 ﷐(𝑟 + 1)﷮𝑡ℎ﷯ 𝑡𝑒𝑟𝑚﷯ = ﷐7﷮42﷯ ﷐﷐𝑛!﷮﷐𝑟−1﷯![𝑛−﷐𝑟−1)﷯!﷯﷮﷐𝑛!﷮𝑟! ﷐𝑛−𝑟﷯!﷯﷯ = ﷐7﷮42﷯ ﷐𝑛!﷮﷐𝑟−1﷯!(𝑛−𝑟+1)!﷯ × ﷐𝑟! ﷐𝑛−𝑟﷯! ﷮𝑛!﷯ = ﷐1﷮6﷯ ﷐𝑛! × 𝑟 × ﷐𝑟−1﷯!﷐𝑛−𝑟﷯!﷮𝑛!﷐𝑟 − 1﷯! (𝑛 − 𝑟 +1)!﷯ = ﷐1﷮6﷯ ﷐𝑟﷐𝑛−𝑟﷯!﷮(𝑛 − 𝑟 +1)!﷯ = ﷐1﷮6﷯ ﷐𝑟 (𝑛−𝑟)!﷮(𝑛 − 𝑟 +1) (𝑛−𝑟)!﷯ = ﷐1﷮6﷯ ﷐𝑟﷮𝑛+1−𝑟﷯ = ﷐1﷮6﷯ 6r = n + 1 – r n – 7r + 1 = 0 Now we have n – 8r + 9 = 0 …(1) n – 7r + 1 = 0 …(2) From (1) n – 8r + 9 = 0 n = 8r – 9 Putting n = 8r – 9 in (2) (8r – 9) – 7r + 1 = 0 8r – 9 – 7r + 1 = 0 r – 8 = 0 r = 8 Putting value of r in (1) n – 8r + 9 = 0 n – 8(8) + 9 = 0 n – 64 + 9 = 0 n – 55 = 0 n = 55 Hence n = 55

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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