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Chapter 9: Problem 6

For what values of \(x\), the numbers \(-\frac{2}{7}, x,-\frac{7}{2}\) are in G.P.?

Short Answer

Expert verified
The value of x for which the numbers form a G.P. is \( x = 1 \) or \( x = -1 \) .

Step by step solution

01

Understand the problem

The problem is asking for what values of x will make the numbers \( -\frac{2}{7}, x, -\frac{7}{2} \) form a geometric progression (G.P.), which is a sequence where each term after the first is found by multiplying the previous one by a constant called the common ratio.
02

Write down the definition of G.P.

For these three numbers to be in G.P., it must be true that \( \frac{x}{-\frac{2}{7}} = \frac{-\frac{7}{2}}{x} \) because a constant ratio must exist between consecutive terms.
03

Cross multiply to solve for x

Cross multiply the equation to solve for x: \( x \cdot x = -\frac{2}{7} \cdot -\frac{7}{2} \) which simplifies to \( x^2 = \frac{2}{7} \cdot \frac{7}{2} \) and then to \( x^2 = 1 \).
04

Find the square root of both sides

Take the square root of each side of the equation to solve for x: \( x = \pm\sqrt{1} \).
05

State the answer

The square root of 1 is 1. Therefore, \( x \) can be either 1 or -1 for the given numbers to form a G.P.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

G.P. definition
A Geometric Progression (G.P.) is a sequence of numbers where each term after the first is generated by multiplying the previous term by a fixed, non-zero number known as the common ratio. This sequence forms a pattern where the product of each term and a consistent factor gives the next term in the line.

For example, in a sequence like 2, 6, 18, 54, ..., the first term is 2, and each term is obtained by multiplying the previous one by 3, which is the common ratio here. Understanding this fundamental concept of G.P. is essential in solving problems related to sequences and series in algebra and higher mathematics.
common ratio
The common ratio in a G.P. is the factor by which consecutive terms are multiplied to obtain the next term. It remains constant throughout the sequence and can be positive or negative, but never zero. To find the common ratio, we simply divide any term in the G.P. by the term preceding it.

In the context of our exercise, we are examining a three-term geometric sequence. To find the value of the middle term, we equate the ratio of the middle term to the first term with the ratio of the third term to the middle term. This common ratio allows the sequence to maintain its geometric nature. Whether the common ratio is a fraction, an integer, or an irrational number, it's the key to generating every term in the sequence after the first.
cross multiplication
Cross multiplication is a technique used to solve equations involving two fractions set equal to each other. It involves multiplying the numerator of each fraction by the denominator of the other fraction and setting the resulting products equal to each other. This method simplifies the process of finding unknown variables in proportion-related problems.

In the exercise, we applied cross multiplication to determine the value of 'x' that would sustain the property of a G.P. By systematically applying the cross-multiplication method, we arrived at an equation that we could solve to find the correct values for 'x'. This is a common and powerful technique when dealing with ratios, proportions, and sequences.

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Most popular questions from this chapter

Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: $$ a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n \geq 2 $$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose \(n^{\text {th }}\) terms are: $$ a_{n}=\frac{n(n-2)}{n+3} ; a_{20} $$

Find the indicated terms in each of the sequences in Exercises 7 to 10 whose \(n^{\text {th }}\) terms are: $$ a_{n}=4 n-3 ; a_{17}, a_{24} $$

If the sum of \(n\) terms of an A.P. is \(\left(p n+q n^{2}\right)\), where \(p\) and \(q\) are constants, find the common difference.

Write the first five terms of each of the sequences in Exercises 1 to 6 whose \(n^{\text {th }}\) terms are: $$ a_{n}=n(n+2) $$
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