Please help with part a Show for all integers n> 1 that 7 1 (34n+ - 5?n-1 ).а.b.Suppose that {a, }is a sequence of real numbers defined recursively byа, 3D1, а, — 2, аnd a,2 %3Vn 21 Write out the first 13 terms and give a closed form formula for a,'n+24n? +14n+1Let P(n):9+13+...+(4n+5) =с.2Show P(k)→ P(k +1). Is P(n) true Vintegers n21?

Name: Please help with part a Show for all integers n> 1 that 7 1 (34n+ - 5?
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please help with part a Show for all integers n> 1 that 7 1 (34n+ - 5?n-1 ).а.b.Suppose that {a, }is a sequence of real numbers defined recursively byа, 3D1, а, — 2, аnd a,2 %3Vn 21 Write out the first 13 terms and give a closed form formula for a,'n

a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For n=1, 34n+1-52n-1=34+1-52-1=35-51=243-5=238 Since 7 divides 238, the statement 734n+1-52n-1 is true for n=1.Inductive step: Assume that the statement is true for n=k. That is, 734k+1-52k-1, where k≥1. Now prove the statement is true for n=k+1 as shown below. 34k+1+1-52k+1-1=34k+4+1-52k+2-1=34k+1·34-52k-1·52=81·34k+1-25·52k-1=56+25·34k+1-25·52k-1=56·34k+1+25·34k+1-25·52k-1=78·34k+1+2534k+1-52k-1Here, 7 divides the term 78·34k+1 .a Since 734k+1-52k-1 by assumption, we have 7 divides the term 2534k+1-52k-1. Therefore, 7 divides 78·34k+1+2534k+1-52k-1. That implies, 734k+1+1-52k+1-1. Hence, the statement is true for n=k+1. Therefore, by induction hypothesis, 734n+1-52n-1 for all integers n≥1.