MATHS PUZZLE # 3 FIRST LEG

MATHS PUZZLE # 3 FIRST LEG --Second Year of Project--
(from the 18th to the 25th of October)

In how many ways can you write 111 as a sum of three whole numbers (integers) which are in geometric progression?

Remember that geometric progression is a series of numbers in which each number is multiplied by a fixed amount (called ratio) in order to get the next number: for example: 1, 5, 25, 125.

You can send the solution to FernandoEscuin@iescampanar.com

SOLUTION

The sum is like this:

 a + ar +ar²  =  a(1 + r + r²) = 111

The factorial decomposition of 111 is:

111 = 1x3x37

We have the following possibilities:

a = 1

1 + r + r² = 111

 r² + r – 110 = 0. The values of r are: 10, -11

With solutions: 1, 10, 100   and  1, -11, 121

a = 3

1 + r + r² = 37

r² + r – 36 = 0, this equation doesn’t have integer solution.

a = 37

1 + r + r² = 33

r² + r – 2 = 0, with roots r=-2, 1

Now, we have new solutions: 37, -74, 148 and 37, 37, 37