Consider the following recurrence relation: a_n=2 cdot a_n-1-3 with a_1=5

remolatg 2021-02-09 Answered

Consider the following recurrence relation:
an=2an13 with a1=5

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Expert Answer

Laaibah Pitt
Answered 2021-02-10 Author has 98 answers
Let the given recurrence relation as follows:
an,=2an13 with a1=5
Let us consider the statement, for all natural number n, as follows
P(n):an,=2n+3,nl
We have to prove the statement P(n) by using Mathematical induction on n.
Show that the statement P(n) holds for n=1.
P(1) is easily seen to be true:
a1=5=21+3
Inductive Step: Show that if P(k) holds then P (k+1) holds.
Assume P(k) holds, that means,
ak=2k+3
Now we must have to show P(k+1) holds, that is, to show
ak+1=2k+1+3
From the recurrece relation we have
ak+1=2ak3 [As given]
=2(2k+3)3 [Induction Hypothesis]
=2ak+233
=2ak+3
Therefore, indeed, P(k+1) is true.
Hence by mathematical induction, we conclude that P(n) is true for all
n1.
an=2n+3,n1
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