Answer to Question #184165 in Discrete Mathematics for Angelie Suarez

Question #184165

MATHEMATICAL INDUCTION AND RECURRENCE


Solve the following. (10 pts each)

1. Prove P(n) = n2 (n + 1)

2. Recurrence relation an = 2n with the initial term a1 = 2.


1
Expert's answer
2021-05-07T09:35:08-0400

1. Given statement is-

"P(n) = n^2 (n + 1)"


at "n=1, P(1)=1(1+1)=2" , Which is even


P(1) is true.

Let us assume that P(k) is true for some positive integer k,

"P(k)=k^2(k+1) \\text{ is even } ~~~~~~~~-(1)"


Now put "k=k+1"

"P(k+1)=(k+1)^2(k+2)"

"=(k^2+1+2k)(k+2)\\\\=k^3+4k^2+4k+2\\\\ =k^3+k^2+3k^2+4k+2"


As "k^3+k^2" is even so,

P(k+1) is true

Hence Given statements is true for all "n\\in N" .


2."a_n=2n, a_1=2"


at "n=2, a_2=4"

at "n=3, a_3=6"

at "n=4, a_4=8"

The Required sequence is- "2,4,6,8,...."

The required recurrence relation is

"a_n=a_n-a_{n-1}," Where "n\\ge 2" and "a_1=2"


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