A sequence is defined recursively by an4an 9 and a13 Find an

A sequence is defined recursively by an+-4an -9 and a1-3. Find an explicit formula for an an Use mathematical induction to prove that the explicit formula is true. Let P(n) denote the statement an +1 = 4an_9. P(1) is the statement that a 1+1 = 3, Then, we have the following a1+1 = 4a1-3 =4( 3 Thus, P1) is true. Assume that P(k) is true. Thus, our induction hypothesis is ak+i-4k-9. We want to use this to show that P(k +1) is true. a(k + 1) +1=4ak + 1-9 = 4(4ak 16ak = 16(3)- X-9 Enter an exact number Therefore, a(k +1)+1 above steps, we conclude by the Principle of Mathematical Induction that P(n) is true for all natural numbers n by the induction hypothesis. Thus, P(k +1) follows from P(k), and this completes the induction step. Having proven the

Solution

a₁ = 3

P(n) ========> a_n+1 = 4a_n – 9

Now P(1) =====> a₂ = 4a1 – 9 = 4*3 – 9 = 3

Now P(k) =======> a_k+1 = 4a_k – 9

P(k + 1) =======> a_(k+1)+1 = 4a_k+1 – 9 = 4[4a_k – 9] – 9 = 16a_k – 36 – 9 = 16*3 – 45 = 3