Practice Prove 3^(k + 1) - 1

MarkFL

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#2
Assuming \(k\in\mathbb{N}\) we may write:

\(\displaystyle 3^{k+1}-1=3(2+1)^{k}-1=3\sum_{j=0}^k\left({k \choose j}2^{j}\right)-1=2\left(3\sum_{j=1}^k\left({k \choose j}2^{j-1}\right)+1\right)\)
 

MarkFL

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#4
Wow! Is there anything in math you don't know?
Isaac Newton once said:

"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

In that analogy, I am far beyond the sand dunes, and if I squint my eyes just right, I can see a boy playing on the shore. :)
 

harpazo

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#5
You certainly fight your way to the right answer. I see this in all your replies.

1. How do you answer math questions when pressured by time (standardized exams)?

2. How fast were you able to complete classroom exams? I do not do well under pressure. The ASVAB, however, was too easy.

3. I can pass the ASVAB today at 53. How about you?

4. Be honest with me. WHY ON EARTH are you not working as a paid math tutor?
 

MarkFL

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#6
1. & 2) I never had any issues with time on exams. I was always one of the first to complete exams, and I always double checked all problems.

3.) I suspect I would do better at the ASVAB now than when I was 17.

4.) As I've said multiple times, I choose not to.