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A sequence `b_(0),b_(1),b_(2), . . .` is defined by letting `b_(0)=5` and `b_(k)=4+b_(k-1)`, for all natural number k. Show that `b_(n)=5+4n`, for all natural number n using mathematical induction.

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We have a sequence `b_(0), b_(1), b_(2),"……."` is defined by lattin g `b_(0) = 5` and `b_(k) = 4 + b_(k-1)`, for all natural numbers k.
Let `P(n) : b_(n) = 5+ 4n`, for all natural numbers n
For `n = 1`
`b_(1) = 5 + 4 xx 1 = 9`
Also, `b_(0) = 5`
`:. b_(1) = 4 + b_(0) = 4 + 5 = 9`
Thus, `P(1)` is true.
Now, assume that `P(n)` is true for `n =k"....."(1)`
Now, to aprove `P(k+1)` is true, we have to show that
`P(k+1) : b_(k+1) = 5 + 4(k +1)`
`b_(k+1) = 4 + b_(k+1-1)` , (as `b_(k) = 4+b_(k-1)`)
` = 4+ b_(k)`
`= 4 + 5 + 4k = 5 + 4(k+1)`
Hence, `P(k+1)` is true whenever `P(k)` is true.
So, by the principle of mathematical induction `P(n)` is true for any natural number n.
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