Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.
4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = (4(4n+1)(8n+7))/6

Answer 1

Can you prove the case when n=1?

Answer 2

@mathmate no, i honestly do not understand how to do :(

Answer 3

The process of mathematical induction is a three step process:
1. prove a base case, usually when n=0 or n=1.
2. ASSUMING that the conjecture is true for the general case n, prove that it is true for the case n+1.
3. If 1 and 2 are both true, then the conjecture is true for all n where \( n\in N \).

Answer 4

@mathmate do you use the equation right of the equal sign to figure out if it is true?

Answer 5

The given conjecture is
4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = (4(4n+1)(8n+7))/6
should be true for all positive integers.
So let's do the first step of trying n=1.
Can you do that?

Answer 6

You will need to evaluate the left-hand side and right-hand side according to the equation.
The left-hand side has n terms, while the right hand side has only one term.

Answer 7

For n=1, both sides have only one term.

Answer 8

@mathmate oh so the left and right equations have to equal each other?

Answer 9

If the conjecture is true, they should be equal. If they are not equal for ANY value of n, the conjecture is not true.

Answer 10

Can you start by evaluating the left-hand side for n=1?

Answer 11

@mathmate oh okay! that makes a lot more sense! so right now when i plug in 1 on each side , i have \[[4(1)][4(1)+2]=\frac{ 4[4(1)+1][8(1)+7] }{ 6 }\]
the left side equals \[4\times6\] or 24
and the right side equals
\[(4\times5\times15)\div6\]
\[300\div6\] which equals 50
so they do not equal each other

Answer 12

And the conclusion is......
But before we go further, I would always check if the question was posted or understood correctly. Teachers don't usually give a freebie like that.

Answer 13

@mathmate well since it is false, i have to show why and how

Answer 14

You already showed a counter example that the conjecture is false for n=1. So the conjecture is false. (Note that, on the contrary, showing that it works for n=1 does not prove that it works for all n).
However, as I mentioned, make sure there are no typos.

Answer 15

... no typos in the question that you posted.

Answer 16

@mathmate i double checked and everything is correct. should i use another number for show that it is false?

Answer 17

In mathematics, if you showed one single case (ex. n=1) is false, the conjecture is false. Compare with murder cases before a judge. Often when the prosecutor proves that the witness has lied about a detail in the testimony, the testimony is thrown out almost all the time.

Answer 18

@mathmate thank you so much for your help , honestly. i finally understand this!! thank you again :) have a great new year!!

Answer 19

yw! :)