Question

Make a conjecture about 8Tn + 1 and then prove it.
Triangular numbers Tn. Hint, try several examples using numbers to get a pattern and make a conjecture

Answer 1

Well you could go that 8Tn + 1 is odd for all n

Answer 2

The formula for a triangular number is n * (n+1) /2
So the 4th triangular number equals 4 * (4+1) / 2
= 20 / 2
equals 10

Answer 3

To prove, clearly all Tn are integers, and so 8Tn must be even. Adding one therefore must give odd.

Answer 4

Hope this is what you are looking for

Answer 5

\[8T{n}+1\]
I am looking for a conjecture dealing with triangular numbers

Answer 6

isnt what i suggested a conjecture then?

Answer 7

Yes but they are meaning another pattern that equals the same such as (2n+1)^2 is the same as 8Tn +1, but I am trying to prove it

Answer 8

Ah why didnt you say so earlier

Answer 9

So you want to prove (2n + 1)^2 = 8Tn + 1?

Answer 10

yes

Answer 11

You know Mathematical Induction?

Answer 12

Actually dw

Answer 13

Ok can we assume Tn = n(n+1)/2? Or do i have to prove that...

Answer 14

Ok I'm gonna assume we can, and prove it later if i get time

Answer 15

I know how to do induction. I can not figure it out at n = k+1

Answer 16

So on the LHS we have (2n+1)^2, and RHS we have 8Tn + 1

work on RHS,
RHS = 8*n*(n+1)/2 + 1
= 4n(n+1) + 1
= 4n^2 + 4n + 1
= (2n+1)^2 expand it if you don't believe me

Answer 17

= LHS

Problem Solved

Answer 18

All good?

Answer 19

May I get the entire induction without the LHS and RHS