Question

Easy question I only need a little help

Answer 1

Use mathematical induction to prove the statement is true for all positive integers n.

10 + 30 + 60 + ... + 10n = 5n(n + 1)

Answer 2

Alright I already hvae it to
\[10(1)=5(1)(1+1)\]\[10=5(2)\]\[10=10\]

Answer 3

Then I changed it to \[10k=5k(k+1)\]

Answer 4

Now I am confused on how to change it to k+1 form. Would that be\[10(k+1)=5(k+1)(k+1+1)?\]

Answer 5

I think it looks good so far.

Answer 6

okay so that is the right form for k+1?

Answer 7

because another one that I did like this was like \[2+4+6+...+2k=k(k+1)\] and then I had to change it to\[2+4+6+...+2k+2(k+1)=(k+1)(k+1+1)\]

Answer 8

then I made it \[k(k+1)+2(k+1)=(k+1)(k+2)\] then I factored and got \[(k+1)(k+2)=(k+1)(k+2)\]

Answer 9

So I'm think maybe \[10+30+60++...+10k+10(k+1)=5(k+1)(k+2)\] and maybe turn that into \[k(k+1)+10(k+1)=5(k+1)(k+2)\]

Answer 10

does that look right to you?

Answer 11

Yes.
5(k + 1) (k + 2) = 5(k + 1)( (k + 1) + 1 )

Answer 12

Since k + 2 is the same as (k + 1) + 1, it's the term after k + 1.
You showed that it works for k + 1. That's how induction works. Good job.
Just remember to show all steps in the usual way.
First prove it works for n = 1
Then assume it works for any integer k.
Then show that is works for k + 1.

Answer 13

alright thank you for your help