Mathematical Induction - Questions and Solutions

Question 1
Important

Question 2

Question 3 Important

Question 4

Question 5 Important

Question 6

Question 7 Important

Question 8 Important

Question 9

Question 10

Question 11 Important

Question 12

Question 13 Important

Question 14

Question 15 Important

Question 16 Important You are here

Question 17 Important

Question 18 Important

Question 19

Question 20

Question 21 Important

Question 22

Question 23 Important

Question 24 Important

Mathematical Induction

Serial order wise

Last updated at April 16, 2024 by Teachoo

Question16 Prove the following by using the principle of mathematical induction for all n ∈ N: 1/1.4 + 1/4.7 + 1/7.10 +…….+ 1/((3𝑛 − 2)(3𝑛 + 1)) = 𝑛/((3𝑛 + 1)) Let P (n) : 1/1.4 + 1/4.7 + 1/7.10 +…….+ 1/((3𝑛 − 2)(3𝑛 + 1)) = 𝑛/((3𝑛 + 1)) For n = 1, L.H.S = 1/1.4 = 1/4 R.H.S = 1/((3(1) + 1)) = 1/((3 + 1)) = 1/4 Hence, L.H.S. = R.H.S , ∴ P(n) is true for n = 1 Assume P(k) is true 1/1.4 + 1/4.7 + 1/7.10 +…….+ 1/((3𝑘 − 2)(3𝑘 + 1)) = 𝑘/((3𝑘 + 1)) We will prove that P(k + 1) is true. R.H.S = ((𝑘 + 1))/((3(𝑘 + 1)+ 1) ) L.H.S = 1/1.4 + 1/4.7 + 1/7.10 +…….+ 1/((3(𝑘 + 1)− 2)(3(𝑘 + 1)+ 1))