Question

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

Answer 1

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

Answer 2

8n + 7 sorry

Answer 3

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

Answer 4

The problem is not clear!!

Answer 5

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

Answer 6

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

Answer 7

yeah thats it

Answer 8

makes a huge difference....

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

Answer 9

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.

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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...

Answer 16

some induction questions use sigma notation

Answer 17

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?

Answer 18

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.

Answer 19

Okay, that makes, thank you so much!!