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

1) 4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = 4(4n+1)(8n+7)/6

2) 1^2 + 4^2 + 7^2 + ... + (3n - 2)^2 = n(6n^2-3n-1)/2

Question

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

1) 4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = 4(4n+1)(8n+7)/6

2) 1^2 + 4^2 + 7^2 + ... + (3n - 2)^2 = n(6n^2-3n-1)/2

anonymous · Answer

i did 2 i just need one now

amistre64 · Answer

if its true for all positive ints, then test it for n=1

amistre64 · Answer

if its true for n=1, then assume its true for all n=k

amistre64 · Answer

im not too sure why "assuming" something to be true actually makes it true, but for some reason thats how this goes

anonymous · Answer

so 4(1)(4(1)+2)/2
= 24/2= 12 not one

anonymous · Answer

so it's not true?

amistre64 · Answer

$$P(1)=4(1)( 4(1) + 2) = \frac{4(4(1)+1)(8(1)+7)}{6}$$
$$P(1)=4\cdot 6 = \frac{10(5)}{1}$$
is your setup correct?

amistre64 · Answer

well, if its setup correctly, and 1 IS a positive integer .... then its not true for n=1

anonymous · Answer

oh duh, yes
and so does that mean you can't do the problem?

amistre64 · Answer

it means that it false since it states that it is true for ALL positive integers

anonymous · Answer

Okay thank you very much :)

amistre64 · Answer

youre welcome