Question

A sequence is defined by the first-order recurrence relation: an=5a_n-1+3 a₀=4

a) Write out the first 5 terms of this sequence.

b) Given that: an=A*5ⁿ+B Show that A=19/4 and B=-3/4.

c) Use mathematical induction to prove that ?_n = 19/4 × 5ⁿ – 3/4

Answer 1

`Hey,

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a)

a(n)=5*a(n-1)+3

Also, a(0)=4

a(1)=5*a(0)+3=5*4+3=23
a(2)=5*a(1)+3=5*23+3=118
a(3)=5*a(2)+3=5*118+3=593
a(4)=5*a(3)+3=5*593+3=2968
a(5)=5*a(4)+3=5*2968+3=14843

b)

Since

a(n)=A*5^n+B

a(1)=5A+B=23---------eq1

a(0)=A+B=4----------eq2

Subtract eq1 and 2

So,

4A=19

A=19/4

SO,

B=4-19/4=-3/4

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