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Let ϕ(x) represent a quadratic polynomial. Given that ϕ(1) equals ϕ(−1) and the terms a₁, a₂, a₃ form an arithmetic progression, then the values ϕ(a₁), ϕ(a₂), ϕ(a₃) will be in which sequence?

1. Arithmetic progression
2. Geometric progression
3. Harmonic progression
4. None of the above

Correct answer: None of the above

Solution

phi(1)=phi(-1) forces no linear term: phi(x)=Ax^2+C. For an AP a-d,a,a+d, phi values are A(a-d)^2+C, Aa^2+C, A(a+d)^2+C; check 2*mid=A(a-d)^2+A(a+d)^2 requires 2a^2=(a-d)^2+(a+d)^2=2a^2+2d^2, false for d!=0. So they are not in AP (nor GP/HP generally): None of the above (idx 3).

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