# Conditional identity consisting of AP and GP

• rama
In summary, you are given that x, y, and z are in geometric progression and a, b, and c are in arithmetic progression. You proved that (xb÷xc)(yc÷ya)(za÷zb)=1 by using the fact that x, y, and z are in GP and a, b, and c are in AP.
rama

## Homework Statement

x,y,z are three terms in GP and a,b,c are three terms in AP
prove that (xb÷xc)(yc÷ya)(za÷zb)=1

## The Attempt at a Solution

(xb-c)(yc-a)(za-b)

since x y z are in GP
xb-c÷yc-a=yc-a÷za-b
(xb- c)(za-b)=yc-a(yc-a)

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rama said:

## Homework Statement

x,y,z are three terms in GP and a,b,c are three terms in AP
prove that (xb÷xc)(yc÷ya)(za÷zb)=1

## The Attempt at a Solution

(xb-c)(yc-a)(za-b)

since x y z are in GP
xb-c÷yc-a=yc-a÷za-b
(xb- c)(za-b)=yc-a(yc-a)

You are given that x, y, and z are in geometric progression. Did you use that fact in your work?

Also, a, b, and c are in arithmetic progression. Did you use that fact in your work?

1 person
got it thank you, I seem to be posting here without thinking hard
next time I won't post without thinking out all options sorry

## 1. What is conditional identity consisting of AP and GP?

Conditional identity consisting of AP and GP is a mathematical concept that states that if two sequences are arithmetic and geometric progressions, respectively, and have the same first term and common difference/ratio, then their sum or product will also form an arithmetic or geometric progression, respectively.

## 2. How is conditional identity consisting of AP and GP useful?

Conditional identity consisting of AP and GP is useful in solving various mathematical problems involving sequences and series. It allows us to find the sum or product of a series without having to manually calculate each term.

## 3. Can conditional identity consisting of AP and GP be used in real-life situations?

Yes, conditional identity consisting of AP and GP can be used in real-life situations, particularly in finance and economics. It is commonly used in calculating compound interest and in analyzing investment growth over time.

## 4. How do you prove conditional identity consisting of AP and GP?

The conditional identity consisting of AP and GP can be proved using mathematical induction. By assuming the identity is true for a specific value of n, we can show that it is also true for n+1, thus proving the identity for all positive integers.

## 5. Are there any limitations to conditional identity consisting of AP and GP?

While conditional identity consisting of AP and GP is a powerful tool in mathematics, it has its limitations. It can only be used for sequences with a common difference/ratio and cannot be applied to divergent sequences. Additionally, it may not be applicable in certain complex mathematical problems.

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