Answer:
The geometric progression is 6,12,24.
Step-by-step explanation:
Given : Three numbers 1, 7 and 19.
To find : What number should be added to each of numbers so that the resulting three numbers form a geometric progression?
Solution :
We know that, In geometric progression the ratio between two number is same.
Let x be the number added in three numbers.
So, The sequence form is 1+x, 7+x, 19+x
As their ratios are same so,
[tex]\frac{7+x}{1+x}=\frac{19+x}{7+x}[/tex]
Solve the expression,
[tex](7+x)(7+x)=(19+x)(1+x)[/tex]
[tex]49+7x+7x+x^2=19+19x+x+x^2[/tex]
[tex]49+14x=19+20x[/tex]
[tex]6x=30[/tex]
[tex]x=5[/tex]
Terms are
[tex]1+x=5+1=6[/tex]
[tex]7+x=7+5=12[/tex]
[tex]19+x=19+5=24[/tex]
Therefore, The geometric progression is 6,12,24.
