1. Chapter 4 Class 11 Mathematical Induction
2. Serial order wise

Transcript

Ex 4.1, 5: Prove the following by using the principle of mathematical induction for all n โ N: 1.3 + 2.32 + 3.33 + โฆ.. n.3n = ((2๐ โ 1) 3^(๐ + 1) + 3 )/4 Let P(n) : 1.3 + 2.32 + 3.33 + โฆ.. n.3n = ((2๐ โ 1) 3^(๐ + 1) + 3 )/4 For n = 1, we have L.H.S =1.3 = 3 R.H.S = ((2.1 โ 1) 3^(1+1) + 3)/4 = (1 ร 3^2 + 3)/4 = (9 + 3)/4 = 12/4 = 3 Hence, L.H.S. = R.H.S โด P(n) is true for n = 1 Assume P(k) is true 1.3 + 2.32 + 3.33 + โฆ.. k.3k = ((2๐ โ 1) 3^(๐ + 1) + 3 )/4 We will prove that P(k + 1) is true. 1.3 + 2.32 + 3.33 + โฆ.. + (k + 1)3k + 1 = ((2(๐ + 1)โ1) 3^((๐+1) + 1) + 3 )/4 1.3 + 2.32 + 3.33 + โฆ.. + (k + 1)3k + 1 = ((2๐ + 2 โ 1) 3^(๐ + 2) + 3 )/4 1.3 + 2.32 + 3.33 + โฆ.. + (k + 1)3k + 1 = ((2๐ + 1) 3^(๐ + 2) + 3 )/4 1.3 + 2.32 + 3.33 + โฆ.. + k3k + (k + 1)3k + 1 = ((2๐ + 1) 3^(๐ + 2) + 3 )/4 We have to prove P(k+1) from P(k) i.e. (2) from (1) From (1) 1.3 + 2.32 + 3.33 + โฆ.. k.3k = ((2๐ โ 1) 3^(๐ + 1) + 3 )/4 Adding (k + 1) 3k+1both sides 1.3 + 2.32 + 3.33 + โฆ.. k.3k + (k + 1)3k + 1 = ((2๐ โ 1) 3^(๐ + 1) + 3 )/4 + (k + 1)3k + 1 = ((2๐ โ 1) 3^(๐ + 1) + 3 + 4(๐ + 1) 3^(๐+1))/4 = ((2๐ใ (3ใ^(๐ + 1)) โ 3^(๐ + 1) ) + 3 + 4(๐)3^(๐+1)+4(3^(๐+1) ))/4 = (2๐ใ (3ใ^(๐ + 1)) + 4(๐) 3^(๐+1) โ 3^(๐ + 1) + 4(3^(๐+1) ) + 3)/4 = (6๐ใ (3ใ^(๐ + 1)) + 3(3^(๐+1) ) + 3)/4 = ((3^(๐+1) )(6๐ + 3) + 3)/4 = ((3^(๐+1) )3(2๐ + 1) + 3)/4 = ((3^1 ร 3^(๐+1) )(2๐ + 1) + 3)/4 = ((3^(๐+2) )(2๐ + 1) + 3)/4 Thus, 1.3 + 2.32 + 3.33 + โฆ.. + k3k + (k + 1)3k + 1 = ((2๐ + 1) 3^(๐ + 2) + 3 )/4 which is the same as P(k + 1) โด P(k + 1) is true whenever P(k) is true. โดBy the principle of mathematical induction, P(n) is true for n, where n is a natural number