SOLUTION: In the sequence 1,4,5,6,7,8,10,16,18,... each number after the first two is the next number that can be expressed as the sum of two previous numbers in only one way. For example, 1

Question 1151381: In the sequence 1,4,5,6,7,8,10,16,18,... each number after the first two is the next number that can be expressed as the sum of two previous numbers in only one way. For example, 10 is included because 6+4=10, and there is no other sum of two other numbers in the sequence that equals 10. The number 15 is not included because both 10+5 and 8+7 equals 15. In a similar sequence that begins with the two numbers 100 and 101, what is the sum of the first ten terms?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

There is no mathematical magic for solving this problem. Simply follow the rules, looking for patterns that make the task of finding the sequence of numbers easier.

The first two numbers are

100, 101,...

Obviously the next number in the sequence is the sum of those first two:

100, 101, 201,...

There are clearly no more numbers in the 200s; what about the 300s? For those, we need to add one number in the 100s and one in the 200s. Since we only have one number in the 200s, it can be added to either of the numbers in the 100s to get other numbers in the sequence:

100, 101, 201, 301, 302,...

If you are paying attention to what kinds of numbers you are getting, you should recognize that it will be quite a while before we get another number with units digit 0. Furthermore, it should be clear that with 100 the only number with units digit 0, all of the numbers 201, 301, 401, 501, ... will be in the sequence:

100, 101, 201, 301, 302, 401, ???, 501, ???, 601, ???, 701......

Now let's go back to an ordered method for finding the terms of the sequence.

We have one number in the 400s; are there others?

402 can be made; but it can be made in two different ways -- 100+302 or 101+301.

403 can be made (101+302); and since we currently have only one number in the list with units digit 2, we won't be able to make 403 in any other way. So

100, 101, 201, 301, 302, 401, 403, 501, ...

There are the first 8 numbers in the sequence. I'll let you take it from there to find the final answer to the question.

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