Express the number 220 as the sum of four numbers that form a geometric progression such that the third term is greater than the first by 44.

Step-by-step explanation:

The first four terms of geometric series is:

Since, we have given information that the third number is greater than 44 that means:

Above equation can be rewritten as:

Now, using:

Here, a=r,b=1

       (1)

The sum of first four terms is:

     (2)

Divide equation (2) by (1) we get:

CASE1: When r=2 in

CASE2:When r=3 in

The series becomes:

From CASE1:

From CASE2:

A geometric progression is characterized by a common ratio.

The terms of the progression are: or

The given parameters are:

The third term is represented as:

The first term is:

So, we have:

Subtract a from both sides

Factor out a

Express as difference of two squares

Make r + 1 the subject

Recall that:

This gives:

Factor out a

Factor out r^2

Rewrite as:

Factor out 1 + r

Substitute in

Divide both sides by 44

Multiply through by r - 1

Collect like terms

Expand

Factorize

Factor out r -2

So, we have:

Calculate a using:

When r = 3

When r = 2

The nth term of a GP is:

So, the terms of the progression are:

or

Read more about geometric progressions at:

link

hello :

70y^8 + 30y^6 = 2 ( 35 y^8 + 15y^6)

the lenght is : 35 y^8

the width is : 15y^6