One could derive the answer from the first sentence.

It must be a level 1 problem.

you need to see the number in first sentence

Oops the answer appeared in the problem already! See Damiann's solution!

Actually, this problem is supposed to determine the three integers but I don't know how to ask for an integer answer so it ends up being a troll. I am very sorry and the supposed solution is as below:

Tools needed to solve the problem:

For three terms $a,b,c$ ,

• If they form A.P, then $2b=a+c$

• If they form G.P, then $b^2=ac$

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Let the three integers be $a,b,c$ . From the given details, we have:

$b^2=ac$

$a+b+c=65$

$2b=(a-1)+(c-19)$

From the second and third equation, we will have

$b=15$

Substitute to the first and second equation, we will have

$a+c=50$

$ac=225$

From here we can construct an equation with the roots $a,c$ ,

$x^2-(a+c)x+ac=0$

$x=5$ or $x=45$

Hence,

$a+b+c=65$