Henry had his head buried deep in his hands, fully frustrated by his Maths homework for the evening. Gretchen walked in to ask what was wrong, but before she got her question out, Henry launched into a tirade.
“It’s completely unfair! How in the world could Mr. Simkin know what Fibonacci would have intended for the rest of his sequence?! Everything I find on-line says that the Fibonacci sequence starts with two 1’s.”
Gretchen had usually been able to help in situations like these, so she calmly asked her brother to explain.
He replied, “Our new Maths teacher, Mr. Simkin, says that it is only a piece of the Fibonacci sequence we are looking at when it is written as 1, 1, 2, 3, 5, 8. Each number is the sum of the previous two, but he argues that the sequence should read …1, 1, 2, 3, 5, 8. The only question we have for homework tonight is, what does ‘a’ equal in the following trail of numbers? …a, b, c, d, 1, 1, 2, 3, 5, 8…”
A smile spread across Gretchen’s face. “You can do this,” she reassured her brother, “and I will be happy to help.”
What does ‘a’ equal in that sequence?
Answer
‘a’ equals 2.
Since each number is the sum of the preceding two, d must equal 0 so that the second 1 may equal 0 plus 1. That means c has to equal 1 so that the first 1 may make sense. Therefore, b equals -1, making 0 the sum of the previous two numbers. So, ‘a’ must equal 2, leaving 1 as the sum of -1 and 2.
Alternatively, you can look at the sequence as subtractions going from right to left:
3 – 2 = 1
2 – 1 = 1
1 – 1 = 0
1 – 0 = 1
0 – 1 = -1
1 – (-1) = 2
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