Question

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

Answer 1

Given; A sequence b0, b1, b2 ... is defined by letting b0 = 5 and bk = 4 + bk – 1 for all natural numbers k.

⇒ b1 = 4 + b0

= 4 + 5 = 9

= 5 + 4.1

⇒ b2 = 4 + b1

= 4 + 9

= 13

= 5 + 4.2

⇒ b3 = 4 + b2

= 4 + 13

= 17

= 5 + 4.3

Let bm = 4 + bm-1 = 5 + 4m be true.

⇒ bm+1 = 4 + bm+1-1

= 4 + bm

= 4 + 5 + 4m

= 5 + 4(m+1)

⇒ bm+1 is true when bm is true.

∴ By Mathematical Induction bn = 5 + 4n is true for all natural numbers n.