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If a, b, c, d form a geometric progression, then which statement is true?

1. a + b, b + c, and c + d are in geometric progression
2. (b − c)² + (c − a)² + (d − b)² = (a − d)²
3. (a² + b² + c²)(b² + c² + d²) = (ab + bc + cd)²
4. Every statement given is correct

Correct answer: Every statement given is correct

Solution

With a,b,c,d=a,ar,ar^2,ar^3: (A) a+b,b+c,c+d have common ratio r -> GP; (B) (b-c)^2+(c-a)^2+(d-b)^2 = (a-d)^2 holds (e.g. 1,2,4,8 gives 49=49); (C) (a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2 holds. So every statement is correct.

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