Thread: Mathematical Induction Arithmetic

I was working on proving using mathematical induction the premise:

P(n): 2 - 2*7 + 2*7^2 - ... + 2(-7)^n = (1-(-7)^(n+1))/4 while n>=0

I got to the point of proving P(n+1),

[1-(-7)^(n+1) + 8(-7)^(n+1)]/4

but I'm not sure arithmetically how that will simplify into the expected outcome of:

[1-(-7)^(n+2)]/4

???

Am I even right that it will simplify to that seeing as how that is what I was expecting out of the original P(n).

Originally Posted by buddyp450

I was working on proving using mathematical induction the premise:

P(n): 2 - 2*7 + 2*7^2 - ... + 2(-7)^n = (1-(-7)^(n+1))/4 while n>=0

I got to the point of proving P(n+1),

[1-(-7)^(n+1) + 8(-7)^(n+1)]/4

but I'm not sure arithmetically how that will simplify into the expected outcome of:

[1-(-7)^(n+2)]/4

???

Am I even right that it will simplify to that seeing as how that is what I was expecting out of the original P(n).

??

If so, then



Check:



Hence, if true for some n, then definately true for n+1 because of that,
so the dominoes are all stacked up.

True for n=0 ?

?

Yes...... Proven

Originally Posted by buddyp450

I was working on proving using mathematical induction the premise:

P(n): 2 - 2*7 + 2*7^2 - ... + 2(-7)^n = (1-(-7)^(n+1))/4 while n>=0

I got to the point of proving P(n+1),

[1-(-7)^(n+1) + 8(-7)^(n+1)]/4 <--this is correct

but I'm not sure arithmetically how that will simplify into the expected outcome of:

[1-(-7)^(n+2)]/4

???

Am I even right that it will simplify to that seeing as how that is what I was expecting out of the original P(n).

Working with the expression , the tricky part here is the application of Distributive Law to a negative term. Now the positive integer 8 becomes negative.

, which is ,