Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
TurtleMatt
Group Title
Prove that each statement is true for all positive integers.
2 + 4 + 6 + ... + 2n = n^2 + n
 2 years ago
 2 years ago
TurtleMatt Group Title
Prove that each statement is true for all positive integers. 2 + 4 + 6 + ... + 2n = n^2 + n
 2 years ago
 2 years ago

This Question is Closed

TurtleMatt Group TitleBest ResponseYou've already chosen the best response.0
What's the first step? What do I add to both sides? I never know what to add for these questions.
 2 years ago

m_charron2 Group TitleBest ResponseYou've already chosen the best response.0
I would start by simply checking what the lefthand side sums up to. Let's take smaller sums before going up to 2n. 2+4+6 = 4*3, right? 2+4+6+8=5*4. 2+4+6+8+10 = 6*5. See a trend here?
 2 years ago

TurtleMatt Group TitleBest ResponseYou've already chosen the best response.0
What do I do with that trend?
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
it is true for n=1 2=1^2+1 ... Assume it is true for n=k so we have 2+4+6+...+2k=k^2+k Now we want to show it is true for n=k+1 2+4+6+...+2k+2(k+1)=k^2+2k+2(k+1) =k^2+2k+2k+2 =k^2+2k+1+2k+1 do you think you can finish now?
 2 years ago

TurtleMatt Group TitleBest ResponseYou've already chosen the best response.0
Yeah, I can. But how did you know to add 2(k+1) to both sides? That's what confuses me.
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
We want to show it is true for n=k+1 right? so we have \[2+2(2)+2(3)+...+2(k+1)\] I replace n with k+1
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
the integer before k+1 is k
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
\[2(1)+2(2)+2(3)+...+2(k)+2(k+1)\]
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
But we know from assuming n=k we had \[2(1)+2(2)+2(3)+...+2(k)=k^2+k\]
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
So I replace 2(1)+2(3)+2(3)+...+2(k) with k^2+k in 2(1)+2(2)+2(3)+...+2(k)+2(k+1) k^2+k +2(k+1)
 2 years ago

freckles Group TitleBest ResponseYou've already chosen the best response.1
I did make a type0 earlier but anyways this last post is good so we have k^2+k+2k+2 k^2+2k+1+k+1 (k+1)^2+(k+1)
 2 years ago

TurtleMatt Group TitleBest ResponseYou've already chosen the best response.0
Thank you!
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.