Q49 of 66 Page 12

Using principle of mathematical induction prove that

for all natural numbers n ≥ 2.


Step1: For n=2, P(n):




Therefore, it is true for n=2.


Step2: Let P(n) be true for n=k.



Now, we need to show P(k+1) is true whenever P(k) is true.


P(k+1):









So, it is true for n=k+1, thus by the principle of mathematical induction P(n) is true for all n ≥ 2


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47

A sequence x1, x2, x3, …. is defined by letting x1 = 2 and for all natural numbers k, k ≥ 2. Show that for all nϵN

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A sequence x0, x1, x2, x3, …. is defined by letting x0 = 5 and

xk =4+xk–1 for all natural numbers k. Show that xn = 5 for all


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 50

The distributive law from algebra states that for real numbers

c, a1 and a2, we have c(a1 + a2) = c a1 + ca2


Use this law and mathematical induction to prove that, for all


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State the first principle of mathematical induction.