Question

The ages of four brothers Alex, Britley, Camdon and Dalton are in geometric progression and in increasing order. If the ages of Alex, Briti
and Camdon are x, 2x - 2 and 3x - 3 respectively, what is the age of Dalton?
6
7.5
9
13.5
4X - 4​

Answer 1

Given:

Ages of four brothers Alex, Britley, Camdon and Dalton are in geometric progression and in increasing order.

Age of Alex = $x$

Age of Britley = $2x-2$

Age of Camdon = $3x-3$

To find:

Age of Dalton = ?

Solution:

First of all, let us learn about the property of a Geometric Progression (G.P.).

Every next term is equal to the previous term multiplied by a common ratio.

i.e.

2nd term = 1st term multiplied by common ratio.

3rd term = 2nd term multiplied by common ratio.

and so on.

In other words:

$\text{Common Ratio} = \dfrac{\text{2nd term}}{\text{1st term}}=\dfrac{\text{3rd term}}{\text{2nd term}}\\\Rightarrow \text{(2nd term)}^2 = \text{1st term}\times {\text{3rd term}}$

Here, we are given the terms as:

$x, 2x-2, 3x-3$.

Let us put them in above formula:

$(2x-2)^2=x\times (3x-3)\\\Rightarrow 4x^2+4-8x=3x^2-3x\\\Rightarrow x^{2} -5x+4=0\\\Rightarrow x^{2} -4x-x+4=0\\\Rightarrow x(x -4)-1(x-4)=0\\\Rightarrow (x -1)(x-4)=0\\\Rightarrow x =1, 4$

Let us try $x=1$,

The ages become: 1, 2-2 =0 which is not possible. Age can not be zero because it is given that they are in increasing order.

Let us try $x=4$,

The ages become:

Age of Alex = 4

Age of Britley = $2\times 4-2 =6$

Common ratio = Age of Britley divided by Age of Alex = $\frac{6}{4} = 1.5$

Age of Camdon = $3\times 4-3 =9$

By concept of GP:

Age of Dalton = Age of Camdon multiplied by Common Ratio

Age of Dalton = $9 \times 1.5 = \bold{13.5}$

So, age of Dalton is 13.5