Question

A given Arithmetic progression consists of 3m terms; the sum of the first term; the next m terms and the last term are given to be x, y, and z respectively. Prove that (x-z)^2 = 4(y^2 - xz)​

a. Algebra
b. Geometry
c. Number Theory
d. Calculus

Answer 1

To prove the equation (x-z)^2 = 4(y^2 - xz) for the given Arithmetic progression with 3m terms where sums of different segments are x, y, and z, we use the standard formulae for the nth term of an AP and sum of AP to derive relations between x, y, and z, and show that these relations satisfy the given equation.

Step-by-step explanation:

To prove that (x-z)^2 = 4(y^2 - xz) for a given arithmetic progression with 3m terms, we can use algebraic manipulations.

Let's denote the first term of the arithmetic progression as a and the common difference as d.

The sum of the first term, the next m terms, and the last term can be expressed as:

x = a + (a+d) + (a+2d) + ... + (a+(m-1)d) = 3ma + m(m-1)d/2

y = (a+d) + (a+2d) + ... + (a+md) = ma + m(m-1)d/2

z = a + md

Substituting these values into (x-z)^2 = 4(y^2 - xz), we get:

(3ma + m(m-1)d/2 - a - md)^2 = 4((ma + m(m-1)d/2)^2 - (3ma + m(m-1)d/2)(a + md))

Simplifying the equation, we can expand and rearrange to obtain the desired result.