Please assist in solving this practice problem step-by-step so I understand it. There are two parts so if you only do the first part that's fine

Use iteration to guess an explicit formula for the sequence: Pk: Phi + 2- 3\Problem 1: Use iteration to guess an explicit formula for the sequence: ek = 7ek-1 + 4, for all integers k 2 2, where e1 = 2. Answer to Problem 1: ex=7ek-1+ 4, for all integers k 2 2 e1 = 2 e2 = 7(e1) + 4 = 7(2) + 4 =71-2 + 70.4 [ As, 70 = 1] ea = 7(e2) + 4 =7(71-2+70-4) + 4 =72-2 + 71-4 + 7'.4 e4 = 7 (e3) + 4 =7(72-2 + 71-4 + 70.4) + 4=73.2 + 72-4 + 71-4 + 70.4 es = 7(e4) + 4 = 7(73-2 + 72.4 + 71.4 + 70.4) + 4 =74-2 + 73-4 + 72-4 + 71.4 + 70.4 Guess: en = 7"-1 . 2 + 7"-2 . 4 + ...+ 72.4+71.4+70.4 en = 2 . 70-1 + 4(70-2 + ...+ 72+71+70) =2.70-1 + 4-5' 7' 70-2-1 - 1 = 2. 7-1 + 4 - [by the formula, ) me= miti- 1-0 7-1 "- 1 k-0 = 2. 7 -1 + 4. 70 - 1 6-7 -1+ 2-70-1 - 2 8-7-1 - 2 (Ans) 6 3 3 Problem 2: A sequence e1, ez, ey ... is defined by letting e1 = 2 and ex=7ex-1+ 4, for all integers k 2 2, Show en = 8-71-1 - 2 for each integer n 21 . 3 Proof (by mathematical induction): Let e1, ez, ear ... be the sequence defined by specifying that e1 = 2 and ex= 7ex-1+ 4 for every integer k 2 2, and let the property P(n) be the equation: 8-7 -1 - 2 3 for each integer n 2 1. Need to prove that for every integer n 2 1, P(n) is true. Basis step: We show that P(1) is true. We have P(1): e1 =2 8-71-1 - 2 8-70 - 2 And we have, e1 = =9 = 2 3 3 Therefore, P(1) is true. Inductive hypothesis: Let m be any integer with m21, and suppose P(m) is true. 8-7-1 - 2 P(m) = em = 3 Inductive step: 8-7" - 2 We must show that P(m+1) is also true. P(m+1) = my1= 3 The left-hand side of P(m+1) is: em+1 = 7e(m+1)-1 +4 by the recursive definition of the sequence = 7em +4 =7.8.7 1 - 2 12 by substitution from inductive hypothesis and 3 3 by writing a number as a fraction = 8.7-70-1- 14 + 12 3 = 8-7"- 2 by the property of exponents and algebra which is the right-hand side of P(m+1). Hence the property is true for n = m + 1. [Since both the basis step and the inductive step have been proved, P(n) is true for all integers n21.]