A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

unicwaan

  • one year ago

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(n + 1)(8n + ) divided by 6

  • This Question is Closed
  1. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    there seems to be something missing from the sum (8n + ??)

  2. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    8n + 7 sorry

  3. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ok.... so prove its true for n = 1 using the general term on the left you get 4 x 6 so the 1st term is divisible by 6 the sum of 1 term is( the right side) gives 4(1 + 1)(8x1 + 7) = 4 x 2 x 15 = 120 which seems to be a huge issue...1st term is 24 but the sum of 1 term is 120... you might like to check my work....but it seems you already have a problem

  4. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The problem is not clear!!

  5. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    and if you look at the sequence 5 x 7 = 35 not dividible by 6 so the sum of the 1st 2 terms won't be divible by 6

  6. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    To me, I understand it as \(4*6 +5*7 +6*8 +.......+4n(n+2)=\dfrac{4(n+1)(8n+7)}{6}\) and prove it by induction

  7. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah thats it

  8. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    makes a huge difference.... so test it with n = 1 what do you get on the left and right side...?

  9. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    But to it, the basic step with n=1 is not true since when n=1, the left hand side is \(4*1(4*1+2) = 4*6\) . the first element of the sequence.

  10. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\frac{ 4(4n + 1)(8n + 7)}{ 6 }\] I forgot the other 4... im so sorry

  11. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    ok... so substitute n = 1 what do you get as the sum on the right side of the equation..?

  12. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    while plug in n=1 to the right hand side, we get \(\dfrac{4(4*1+1)(8*1+7)}{6}= 50\)

  13. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    so it seems it fails the initial step of proof by induction

  14. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the right side would equal 50... yeah what @Loser66 just said. and I dont understand why n equals 1

  15. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    I'm sure in the question you are given an initial condition that n = 1, 2, 3, ... but if it's not stated the normal process is to start with n = 1 and prove the left hand side is equal to the right had side so n = 1 is really saying find the 1st term.... and show what the sum of 1 term is...

  16. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    some induction questions use sigma notation

  17. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So if the sum of the left side (when plugging in 1) doesn't equal the first term of the right side, then the statement is false?

  18. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    the left side is the sum of the terms....with the last term being the general case the right side is the general case for the sum of the terms.... so the initial task is to prove the 1st term is equal to the sum of 1 term... it might seem trivial... but it is an essential part of the process in your question things failed the initial test so there is no need to go further.

  19. unicwaan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Okay, that makes, thank you so much!!

  20. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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
  • 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.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.