Question

1. Draw a set of railroad tracks such that the lengths of horizontal ties form a geometric sequence.

Answer 1

1. Draw a set of railroad tracks such that the lengths of horizontal ties form a geometric sequence.

Answer:

A geometric sequence is a sequence whose terms keep a constant ratio, for example, 1, 2, 4, 8. 16, ... As you see the ratio is between a term and the previous one is 2/1 = 4/2 = 8/4 = 16/8 = 2.

With that in mind, you can draw a sequence of rail roads with lenghts 1, 2, 4, 8, 16 ,...

This is it:

-                                     (1)
--                                    (2)
----                                  (4)
--------                             (8)
---------------                    (16)
------------------------------  (32)


2. Write the first four terms of the sequence.

Answer: 1, 2, 4, 8

3. Write a recursive formula for the sequence.

Answer: A(n) = 2 * A(n-1)

That means that the nth terms is equal to the (n-1)th term times 2.

4. Write an explicit formula for the sequence.

Answer:

An = 2 ^ (n - 1), where n is the number of the term = 1, 2, 3, 4, 5, 6, ...

You can verify that that generate:

A(1)  = 2^(1 - 1) = 2^0 = 1

A(2) = 2 ^ (2-1) = 2^1 = 2

A(3) = 2^(3-1) = 2^2 = 4

A(4) = 2^(4-1) = 2^3 = 8