Question

Three numbers x y z are in Ap as well as in GP. What can you say about the numbers?​

Answer 1

Step-by-step explanation:

The full form of A.P. is Arithmetic Progression. An A.P. is a sequence of numbers such that the difference between the consecutive terms is constant. For example 2,3,4 are in A.P. with common difference 1.

What is G.P.?

The full form of G.P. is Geometric Progression. A G.P. is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio. For example, 1,3,9 are in G.P. with common ratio 3.

While going through the problems of A.P., you may come across a term A.M. . Now the full form of A.M. is Arithmetic Mean.

If x,y and z are in A.P. then, y=x+z.

Now, while going through the problems of G.P., you may come across a term G.M.

The full form of G.M. is Geometric Progression.

if x,y,z are in G.P. then y^2 =xz.

We may get this question as,

If a, b & c are in A.P. and G.P.; Prove that a=b=c.

Solution to your question.

Let us consider the numbers as a, b and c.

According to the question; a,b and c are in A.P. and G.P.

Therefore, a+c=2b ————- equation no. 1

b^2= ac ———— equation no.2

Squaring both sides of equation no. 1, we get;

Or, a^2 + 2ac +c^2 = 4(b^2)

Or, a^2 + 2ac+ c^2 = 4ac [ Since, b^2=ac]

Or, a^2 + 2ac -4ac + c^2 = 0

Or, a^2 -2ac + c^2 =0

Or, (a-c)^2=0

Since the square of an expression is zero; hence the value of the expression also must be zero.

Or, a-c=0

Or, a=c ——- equation no. 3

From equation 1 we know,

a+c=2b

Or, a+a=2b [Since, a=c]

Or, 2a = 2b

Or, a=b ——-equation no.4

Therefore, a=b=c.