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  • A sequence b0, b1, b2 ... is defined by letting b0 equalsto 5 and bk equalsto 4 plus bk – 1 for all natural numbers k. Show that bn equalsto 5 plus 4n for all natural number n using mathematical induction.

A sequence b0, b1, b2 ... is defined by letting b_0=5 and b_k = 4 + b_{k -1} for all natural numbers k. Show that for all natural number n b_n = 5 + 4n using mathematical induction.

Answers (1)

Given:

The sequence b0, b1, b2,... if defined if we let, b_0 = 5& for all natural numbers k. b_k = 4 +b_{k-1}Thus,

b_1 = 4 + b_0\\ = 4 + 5\\ = 9 \\ =5 + 4.1

b_2 = 4 + b_1\\ = 4 + 9\\ = 13\\ = 5 + 4.2\\

Now, let us consider,b_m = 4 + b_{m-1} = 5 + 4m to be true.

Thus,    

b_{m+1} = 4 + b_m = 4 + 5+ 4m \\= 5 + 4(m+1)\\

Thus, bm+1 is true if bm is true

Hence, by mathematical induction,

For each natural no. n it is true that, b_n = 5 + 4n

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