To solve this, let's first understand what it means for numbers to form an arithmetic progression (A.P) and a geometric progression (G.P).
Arithmetic Progression (A.P): A sequence of numbers is said to be in arithmetic progression when the difference between any two successive members is a constant. For example, in the sequence
a,b,c, where
b and
c are the successive terms after
a, they must satisfy:
b−a=c−bSimplifying, we get:
2b=a+c Geometric Progression (G.P): A sequence is in geometric progression when each term after the first is multiplied by a constant called the common ratio. In the sequence
a,b,c, they must satisfy:
‌=‌If
a,b, and
c are non-zero, we can rearrange the equation as:
b2=acNow, we know that
a,b,c are both in A.P. and G.P. The key to solving this is to see what happens when we apply the conditions of both progressions. From the G.P. condition,
b2=ac. From the A.P. condition,
2b=a+c. If we substitute
a+c=2b into the G.P. equation:
b2=aâ‹…c Replace
a+c with
2b :
b2=a⋅(2b−a)Let's simplify this:
b2=2ab−a2This actually is a quadratic equation in terms of
a :
a2−2ab+b2=0which simplifies to:
(a−b)2=0Thus,
‌a−b=0‌a=bIf
a=b, then substituting this back in
a+c=2b :
‌a+c=2a‌c=aThus,
a=b=c which concludes that all three numbers must be equal in both A.P. and G.P. when they are non-zero and effective.
Therefore, Option A is correct:
a=b=c