the sum of three number in geometric sequence is 14

Question

the sum of three numbers in geometric sequence is 14. if the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in an arithmetic sequence. find the geometric sequence

Answer 1

Given is

a1+a1*r + a1*r^2=14

 

a1= 14/(1+r+r^2)

 

a1+1,    a1*r+1,  a1*r^2-1  are in arithmetic sequence as given

 

So,

 

a1*r+1-a1-1.    =  a1*r^2-1-a1*r-1

 

simplifying and substituting the value of a1 in terms of r from above

 

We have the expression

 

4*r^2-10r+4=0

 

Solving this quadratic equation

 

r=2   or   1/2

 

The geometric sequence for r = 2

 

2  , 4,  8

 

The geometric sequence for r=1/2

 

8  , 4,    2