Question

Three non-zero real numbers form an A.P. and the squares of these numbers taken in the same order form a G.P. Then the number of all possible common ratio of the G.P. is : (a) 1 (b) 2 (c) 3 (d) none of these

Short answer

Answer: 1.

Step-by-step solution

Write the arithmetic progression

Let the three non-zero real numbers be a-d, a, and a+d, such that a ≠ 0 and a-d ≠ a+d. These numbers are in an arithmetic progression.

Write the geometric progression

Now, the squares of these three numbers are \((a-d)^2\), \(a^2\), and \((a+d)^2\). According to the problem, these are in a geometric progression.

Write the relation between the progressive numbers in G.P.

Since the squares are in a geometric progression, we can write the relation as follows: \[(a^2)/((a-d)^2) = ((a+d)^2)/(a^2)\]

Simplify the equation

Now, we can simplify the equation to get the following: \[(a^2)(a^2 - 2ad + d^2) = (a^2)(a^2 + 2ad + d^2)\] Divide both sides by \((a^2)^2\) to get: \[(1 - 2(ad)/a^2 + (d^2)/a^2) = (1 + 2(ad)/a^2 + (d^2)/a^2)\]

Rearrange the equation and isolate d

Rearrange the equation to isolate d in order to find the possible values of common ratio in the G.P.: \[4(ad)/a^2 = 2(d^2)/a^2\] Cancel out the common terms and divide by 2 to get: \[(ad)/a^2 = (d^2)/a^2\]

Check possible values of d

Now, we have two possible values for d: 1. d = 0 (which is not allowed since the numbers must be non-zero) 2. d = a (which gives us a valid value) So, there is only one solution for the value of d, which is d = a.

Conclusion

Since there is only one valid value of d, there is only one possible common ratio for the G.P. Therefore, the answer is (a) 1.

Key concepts

Arithmetic Progression

In an arithmetic progression (A.P.), sequences of numbers are arranged neatly in such a way that the difference between consecutive terms remains constant. This difference is known as the "common difference." For instance, in the sequence 2, 5, 8, the common difference is 3.

Using algebra to understand A.P., assume the first term is denoted by "a," and the common difference by "d." Therefore, the nth term of A.P. is given by the formula:

• First term: \( a \)
• Second term: \( a + d \)
• Third term: \( a + 2d \)

In the given exercise, we begin with three terms in A.P.: \(a-d\), \(a\), and \(a+d\). The consistent spacing of these terms around a central value makes this a balanced sequence.

Recognizing an arithmetic progression helps in identifying patterns and solving related problems effectively.

Geometric Progression

Unlike an arithmetic progression, a geometric progression (G.P.) requires multiplying each term by a constant factor called the "common ratio" to obtain the next term. Consider the sequence 3, 6, 12; here, the common ratio is 2.

Mathematically speaking, if the first term is "a" and the common ratio is "r," then the nth term of a G.P. follows the pattern:

• First term: \( a \)
• Second term: \( ar \)
• Third term: \( ar^2 \)

In the exercise, the squares of the numbers from the arithmetic progression form a G.P.: \((a-d)^2, a^2, (a+d)^2\). This transformation from A.P. to G.P. reveals how arithmetically spaced numbers adapt to geometric scaling, a key insight for tackling the problem at hand.

Common Ratio

The common ratio in a geometric progression is pivotal as it dictates how rapidly the sequence escalates or diminishes. It is calculated by dividing any term by the preceding term.

In the context of the exercise, the relation given by:\[\frac{a^2}{(a-d)^2} = \frac{(a+d)^2}{a^2}\]helps determine the common ratio. The simplification of this equation eventually reveals that the only consistent and allowable value for the common ratio is 1. This means each term is exactly the square of its arithmetic counterpart, maintaining a perfect geometric relation.

Understanding the power of the common ratio aids in solving similar problems effectively, demonstrating its fundamental role in linking arithmetic and geometric transformations.