Concept: Arithmetic Progression (AP): The series of numbers where the difference of any two consecutive terms is the same, is called an Arithmetic Progression.
If three number a, b and c in AP, then b-a=c-b⇒ 2b=a+c
Geometric Progression (GP): The series of numbers where the ratio of any two consecutive is the same, is called a Geometric Progression
If three numbers a, b and c are in GP, then
b
a
=
c
b
⇒b2=ac
Harmonic Progression (HP): The series of numbers where the reciprocals of the terms are in Arithmetic Progression, is called a Harmonic Progression
If three numbers a, b and c are in HP, then
1
a
+
1
c
=
2
b
Calculation: Ley's say that the second degree polynomial is f(x)=px2+qx+r. From the given information: f(1)=f(−1) ⇒ p(1)2+q(1)+r=p(−1)2+q(−1)+r ⇒ p+q=p−q ⇒ 2q=0 ⇒ q=0 ∴f(x)=px2+0(x)+r
⇒f(x)=px2+r And, f′(x)=2px Now, f′(a)=2pa,f′(b)=2pb and f′(c)=2pc Since a,b and c are in AP, we have: 2b=a+c Multiplying by 2p, we get: (2p)2b=(2p)a+(2p)c ⇒2(2pb)=2pa+2pc ⇒2pa,2pb and 2pc are in AP. ⇒f′(a),f′(b) and f′(c) are in AP