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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
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

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campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2there seems to be something missing from the sum (8n + ??)

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2ok.... 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

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1The problem is not clear!!

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2and 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

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1To 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

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2makes a huge difference.... so test it with n = 1 what do you get on the left and right side...?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1But 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.

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{ 4(4n + 1)(8n + 7)}{ 6 }\] I forgot the other 4... im so sorry

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2ok... so substitute n = 1 what do you get as the sum on the right side of the equation..?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1while plug in n=1 to the right hand side, we get \(\dfrac{4(4*1+1)(8*1+7)}{6}= 50\)

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2so it seems it fails the initial step of proof by induction

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.0the right side would equal 50... yeah what @Loser66 just said. and I dont understand why n equals 1

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2I'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...

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2some induction questions use sigma notation

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.0So 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?

campbell_st
 one year ago
Best ResponseYou've already chosen the best response.2the 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.

unicwaan
 one year ago
Best ResponseYou've already chosen the best response.0Okay, that makes, thank you so much!!
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