Query about an induction proof (1 Viewer)

[vds700]

I have attached the question and solution. In the solution, i cant understand what they did in the line under where it says "On multiplying numerator and denominator by (k + 1)", how the (k + 3) from the line before disappeared and they introduced the (2k - 1).

I appreciate any help.

 

[lyounamu]

I have attached the question and solution. In the solution, i cant understand what they did in the line under where it says "On multiplying numerator and denominator by (k + 1)", how the (k + 3) from the line before disappeared and they introduced the (2k - 1).

I appreciate any help.

They just multiplied (k+1) to the numerator and denominator at the same time.

By doing that, they can separate the equation where you originally proved n=k. And further simplification from then on.

 

[hon1hon2hon3]

althought i dont quite understand all the working out , but look at the first equation . . (k+1)(k+2) . . . . = (something) . . and the next one where they subbed k = k + 1 . . . soo the (k +1) before , became (k + 2 ) . . soo they introduced the (k+1)/(k+1) . . . which is equal to one , and this is what they mean , times by (k+1)/(k+1) soo now the equations looks like (k+1)(k+2)(k+3) . . . = (something) . . . soo then they can used the equation originally.

dont know if this make sense to you . . .

 

[Yamiyo]

The first line contains (k+1)(k+2)(k+3)...2k
The second line contains (k+1)(k+2)...(2k-1)2k

Now these mean exactly the same thing:
(k+1)(k+2)(k+3)(k+4)(k+5)... keep doing this until you get to 2k.
So the (k+3) has just been put into the ...

Same thing with (2k-1), that is, it's just in the ... in the first line.
Here they just stopped at one BEFORE 2k (that is, 2k-1) and then finished it by multiplying by 2k.

 

[lyounamu]

To puy it simply, the reason why they introduce (k+1) to the numerator and the denominator is that, by dong so, they can form an equation where they can relate the equation back to the earlier assumption you made. Even though they introduced (k+1) since they multiplied to the both sides, you are basically multiplying it by 1 which does not alter the equation.

 

[vds700]

The first line contains (k+1)(k+2)(k+3)...2k
The second line contains (k+1)(k+2)...(2k-1)2k

Now these mean exactly the same thing:
(k+1)(k+2)(k+3)(k+4)(k+5)... keep doing this until you get to 2k.
So the (k+3) has just been put into the ...

Same thing with (2k-1), that is, it's just in the ... in the first line.
Here they just stopped at one BEFORE 2k (that is, 2k-1) and then finished it by multiplying by 2k.

ah i get it now. Thanks for the clear explanation