The numbers x, y and z are successive terms of the arithmetic sequence with a common difference of d=4. What is the sum of these numbers, if the numbers x, y and z+8 are successive terms of the geometric sequence?

 The sum of the numbers is 3x + 12

Step-by-step explanation: A brief explanation of an arithmetic progression would be useful for a start.

An arithmetic progression (also known as arithmetic sequence) is series of consecutive numbers in which every term in the entire series is determined by adding a common difference to the previous term. In other words, if x and y are two successive numbers in an arithmetic term, y can be determined by adding the common difference to x. Similarly, the term that comes after y is determined by adding the same common difference to y.

As stated in the question, the numbers x, y and z are successive terms of an arithmetic progression, and the common difference is given as 4. This simply means;

y = x + 4

z = y + 4

z = (x + 4) + 4

z = x + 8

The terms x, y and z can now be re-written as follows,

x, x + 4, and x + 8

The sum of these numbers therefore is derived as,

Sum = x + x + 4 + x + 8

Sum = 3x + 12

the answer is ! ?