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)

Step-by-step explanation:

1.Given

4.6+5.7+6.8+.....+4n(4n+2)=

First , we prove for n=1

= 4.6=24

Left hand side :

=

=24

Right hand side:

=

Put n=1

=

=

=

=50

Hence, left hand side≠ right hand side

Therefore, the given statement is false.

2. =12+42+72+.......(3n-2)2=

similarly , we prove that in the same manner

First , we prove for n=1 the given statement is true or false.

Take n=1

=12

=(3n-2)2

Put n=1 then

=

=2

Put n=1

Then we get

=

=

=2

Hence , left hand side≠right hand side ≠

Therefore, the given statement is false.

3. Given that

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

Put n= 1

=2

Put n=k then we get the statement  Pk

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

Now , Put n= k+1 and then we get the statment Pk+1

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