Q47 of 66 Page 12

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




Step1: For n=1



So, it is true for n=1.


Step2: 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.


More from this chapter

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Prove that the number of subsets of a set containing n distinctelements is 2n for all n ϵ N.

 46

A sequence a1, a2, a3, …... is defined by letting a1 = 3 and ak = 7 ak – 1 for all natural numbers k ≥ 2. Show that an = 3.7 n-1 for all n ϵ N

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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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 49

Using principle of mathematical induction prove that

for all natural numbers n ≥ 2.