Question

PLEASE HELP AND SHOW ALL WORK
7.04

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.
(4 points each.)

1. 4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = quantity four times quantity four n plus one times quantity eight n plus seven divided all divided by six


2. 12 + 42 + 72 + ... + (3n - 2)2 = quantity n times quantity six n squared minus three n minus one all divided by two


For the given statement Pn, write the statements P1, Pk, and Pk+1.
(2 points)

2 + 4 + 6 + . . . + 2n = n(n+1)

Answer 1

1]
4*6+5*7+6*8+.....+4n(4n+2)=4(4n+1)(8n+7)/6
If we choose n=1, then 4*6=24 but 4(4*1+1)(8*1+7)/6=50. This implies that the general trm for the pattern shown is wrong. It should have been (n+3)(n+5) and not 4n(4n+2).

2] 12+42+72+.......+(3n-2)2=n(6n²-3n-1)/2
Let's set n=1, this means that 12=12.
But n(6n²-3n-1)/2
=1(6*1²-3*1-1)/2
=(6-3-1)/2
=2/2
=1
This shows that the general term is incorrect. It should have been (30n-18) which when simplified we get 6(5n-3). Even if we get to correct the left hand side the sequence will still not be equal to what's on the right given n=1.

3] 2+4+6+....+2n=n(n+1)
P(1):2=1(1+1)
P(m):2+4+6+..+2m=m(m+1)
P(m+1):2(k+1)=(k+1)(k+2)

Answer 2

Answer

answer C

Step-by-step explanation: