Question

The condition for which three numbers (a,b,c) are in A.G.P is? thank you

Answer 1

Any (a,b,c) are in arthmetic-geometric progression

Explanation:

Arithmetic geometric progression means that getting from one number to the next involves multiplying by a constant then adding a constant, i.e. if we are at `a`, the next value is
`m cdot a + n` for some given `m, n`.

This means we have formulae for `b` and `c`:

`b = m cdot a + n`
`c = m cdot b + n = m cdot (m cdot a + n) + n = m^2 a + (m+1)n`

If we're given a specific `a`, `b`, and `c`, we can determine `m` and `n`. We take the formula for `b`, solve for `n` and plug that into the equation for `c`:
`n = b - m * a implies c = m^2 a + (m+1)(b - m*a)`
`c= cancel{m^2a} + mb - ma \cancel{- m^2a} + b`
`c = mb - ma + b implies (c-b) = m(b-a) implies m = (b-a)/(c-b)`

Plugging this into the equation for `n`,
`n = b- m*a = b - a * (b-a)/(c-b) = (b(c - b) - a(b-a))/(c-b)`

Therefore, given ANY `a,b,c`, we get exactly find coefficients that will make them an arithmetico-geometric progression.

This can be stated in another way. There are three "degrees of freedom" for any arithmetico-geometric progression: the initial value, the multiplied constant, and the added constant. Therefore, it takes three values exactly to determine what A.G.P. is applicable.

A geometric series, on the other hand, only has two: the ratio and the initial value. This means it takes two values to see exactly what geometric sequence is and that determines everything afterwards.

Answer 2

No such condition.

Explanation:

In an arithmetic geometric progression, we have term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression, such as

`x*y,(x+d)*yr,(x+2d)*yr^2,(x+3d)*yr^3,......`

and then `n^(th)` term is `(x+(n-1)d)yr^((n-1))`

As `x,y,r,d` can all be different four variables

If three terms are `a,b,c` we will have

`x*y=a`; `(x+d)yr=b` and `(x+2d)yr^2=c`

and given three terms and three equations,

solving for four terms is generally not possible and relation depends more on specific values of `x,y,r` and `d`.