Step 1: Use the AP property
Since p,q,r are in Arithmetic Progression (AP), their common difference is the same:
q−p=r−q=d
This means q=p+d and r=p+2d.
Step 2: Use the GP property
We are given that (q−p),(r−q), and p are in Geometric Progression (GP).
Substituting the AP common difference d:
d,d,p are in GP.
For three numbers a,b,c to be in GP, b2=ac. Thus:
d2=d⋅p
Since p,q,r are unequal, d=0. Therefore, we can divide by d:
d=p
Step 3: Express p,q,r in terms of p
p=p
q=p+d=p+p=2p
r=p+2d=p+2p=3p
Step 4: Calculate the required ratio
We need to find the ratio (p+q):(q+r):(r+p):
p+q=p+2p=3p
q+r=2p+3p=5p
r+p=3p+p=4p
The ratio is 3p:5p:4p, which simplifies to 3:5:4.
Conclusion: The correct option is (c) 3:5:4.
