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These are Sum Numbers

There is a sequence of numbers in which every number (except the first two) is the sum of the two numbers before it.

If the first number is 1, and the tenth number is 111, what is the second number?

Be sure to show and explain all of your work. If you use variables, be sure to define them clearly. Please include a check of your answer to verify that it works. An exact answer is needed, not an approximation.

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Bonus: Starting with any two integers, create a sequence using the same process of generating the next number by adding the last two.

Find the sum of the first ten numbers. Compare this sum to eleven times the seventh number in the sequence.

What is the result, and can you prove algebraically that this relationship will always work?

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