Maths induction question, please someone help (1 Viewer)

[matt161989]

the question is:
prove by mathematical induction, 12 to the power of n > 7 to the power of n + 5 to the power of n when n is larger than or equal to 2.
Thank you to anyone who can help me.

 

[airie]

Erm...what exactly is needed to be proven? The second part of the sentence seems to be just the description of a subject, not a complete statement

 

[matt161989]

I have to prove the statement that: 12 to the power of n is larger than 7 to the power of n + 5 to the power of n. when n is larger than or equal to 2

 

[airie]

Does it have to be proven by induction? OK here goes...
When n=2, 12²=144
>7²+5²=74 - base case proven;
Assuming 12ⁿ>7ⁿ+5ⁿ,
12ⁿ⁺¹
=12*12ⁿ
>12*7ⁿ+12*5ⁿ
Since 12>7 and 12>5,
12*7ⁿ>7ⁿ⁺¹, 12*5ⁿ>5ⁿ⁺¹
-->12ⁿ⁺¹>7ⁿ⁺¹+5ⁿ⁺¹
--> Induction complete.

Or, you can just prove it directly for the general case n>2:
12ⁿ
=(7+5)ⁿ
=7ⁿ+n*7^n-1*5+nC2 * 7^n-2*5²+...+n*7*5^n-1+5ⁿ by Binomial Theorem, since n>2
>7ⁿ+5ⁿ
QED.

EDIT: Just a note, I wrote 'n choose 2' as nC2, is that right? I usually use the other notation with the big brackets, btu can't seem to do that here...

 

[matt161989]

thanks for that, it helped heaps, i think i get it now.