SOLUTION: What is the sum of the first 100 consrcutive odd numbers, and do they illustrate inductive or deductive reasons.

You can do it in your head.
        
You know the 100th one has to be 199 because the next even number after each
term is twice the number of terms I have so far. So you want to stop at 199,
because the next higher even number is 200, which is twice 100, the number of
terms I want.  

If you knew the average of all those terms of odd numbers, you'd just have to
multiply that average by the number of terms, 100, to get the same answer that
you'd get if you added took the time and patience to add them all together.

Since the odd numbers are evenly spaced, 1,3,5, etc. through 199, it's
obvious that their average is the same as the average of the just the
first and last terms, (1+199)/2 = 200/2 = 100. [You can do that simple math
in your head].  And there are 100 terms so 100×100 is 10000 (which is also
simple enough to do in your head, so 10000 what you'd get if you added up 
the first 100 odd numbers. 

You can find the sum of any arithmetic sequence by just multiplying the
average of the numbers by the number of numbers.  And you can find that
average by just averaing the first and last terms.

I think that's inductive reasoning.  You might post again just asking that
question.  Maybe somebody else will know and give a second opinion.  I do know
that you prove the formula  INDUCTIVELY.
but the formula is only the above reasoning translated into mathematical
symbols, so you don't have to reason it out every time, you can just plug in the
formula, and like a robot, turn the crank and out pops the answer -- no
reasoning required, except the faith that if you do that you'll end up with
the right answer.  But if you think it through, inductive reasoning is used
here.

Edwin