A large number of birds are sitting on a set of 5 electrical wires as follows.

Question

The five horizontal wires are separated in a vertical plane, and birds are arranged on them according to the following rule: The number of birds on each wire equals twice the total of the birds on the 2 wires immediately below (if there are such wires), plus one bird more.

If there is an even number of birds >0 on the lowest wire, using an index number n = [1, 2, 3, …] write an expression for all possible values of the total number of birds on all five wires combined.

Answer 1

It's most important to build from the proper base: 2n . That way, n=1 handles the case for 2 birds on the lowest wire (the minimum), and each increment of n corresponds to a permitted even number of birds on the bottom wire. Successive wires then hold: 4n+1, 12n+3, 32n+9, and 88n+25, for the sum of 138n+38 . There is no way you would eyeball this result! Common errors are: not adding the “extra” (i.e. the +c) birds before doubling, and misreading the rule, since birds usually congregate on the lower wires in nature! Note: this series is in the Fibonacci class; it's easier to iterate than to try to create general formulas for terms or for their running sum.

You may note that the "initial condition" of 2n birds on the lowest wire overrides the general "propagation" rule, but that's not uncommon in math problems, not everything will be specified in tedious detail. It is, however, important to be alert to possibilities of ranges for variables being other than an obvious interpretation, and for figures being drawn NOT to scale, to deceive you!