1, 3, 4, 7, 11, 18, what next?

Question

Find the next term of 1, 3, 4, 7, 11, 18, ? Then explain how this sequence is related to the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ..., say how fast it grows, and find a surprising exact connection between the two sequences.

Accepted Answer

The next term is 29. Each term is the sum of the previous two; these are the Lucas numbers. The rule. 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18, 11 + 18 = 29, then 47, 76, 123, ... Relation to Fibonacci. The Fibonacci numbers use the same rule with a different start (1, 1). Any sequence with the "add the previous two" rule is a combination of these two, and ratios of consecutive terms in all of them approach the golden ratio, about 1.618: 29/18 = 1.611, 47/29 = 1.621. The surprising connection. Each Lucas number is the sum of the Fibonacci numbers on either side of the corresponding posit…

1, 3, 4, 7, 11, 18, what next?

Your answer

Discussion

Learn the concepts

Related questions