Question

In a GP of positive terms, any term other than the first two terms is equal to the sum of the two preceeding terms. Then, the common ratio of the GP is (a) \(\frac{\sqrt{5}+1}{2}\) (b) \(\frac{-1+\sqrt{5}}{2}\) (c) \(\frac{-1-\sqrt{5}}{2}\) (d) \(\frac{1}{2}\)

Short answer

Answer: (a) \(\frac{\sqrt{5}+1}{2}\)

Step-by-step solution

Setting Up the Equation

In our given GP: $$ar^2 = a + ar.$$ Divide both sides of the equation by \(a\) to isolate the common ratio: $$r^2 = 1 + r.$$

Solving for the Common Ratio

Subtract \(r\) and \(1\) from both sides of the equation to form a quadratic equation: $$r^2 - r - 1 = 0.$$ Now we solve for the roots of the quadratic equation using the quadratic formula: $$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$ where \(a = 1\), \(b = -1\), and \(c = -1\). Substituting these values into the formula, we get: $$r = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}.$$ Solving this expression yields the following: $$r = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}.$$ The two possible values for the common ratio are: $$r_1 = \frac{1 + \sqrt{5}}{2} \text{ and } r_2 = \frac{1 - \sqrt{5}}{2}.$$

Identifying the Positive Common Ratio

Since we are given that the GP has positive terms, we want the positive common ratio. Among the two possible values for the common ratio, \(r_1\) is positive and \(r_2\) is negative. Therefore, the correct common ratio is: $$r = \frac{1 + \sqrt{5}}{2}.$$ Comparing our result to the answer choices, we find that our answer is: (a) \(\frac{\sqrt{5}+1}{2}\).

Key concepts

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 4, 8, 16, ... is a GP where each term is twice the previous term, so the common ratio is 2.

Understanding a GP is crucial in many areas of mathematics and applied sciences. It has unique properties that are leveraged in solving compound interest problems, population growth models, and in the analysis of geometric structures.

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is two. The general form of a quadratic equation is
\( ax^2 + bx + c = 0 \),
where
\( a \),
\( b \), and
\( c \), are constants, and
\( a \) is not zero.

These equations appear frequently in a variety of contexts, including physics, engineering, finance, and many more. They are especially noteworthy in understanding projectile motion, optimization problems, and design of structures. The ability to solve quadratic equations is foundational for higher-level math and science studies.

Common Ratio of GP

The common ratio of a Geometric Progression is the factor by which we multiply each term to get the next term. It is a constant across the series and is represented by the symbol
\( r \).
For instance, in the GP series 3, 6, 12, 24, ..., the common ratio is 6 divided by 3, which is 2. The concept of the common ratio is significant because it defines the property of the GP and determines the behavior of the sequence as it progresses, whether it diverges, converges, or stays constant.

If the common ratio is positive, the terms of the GP will all have the same sign. If it's negative, the terms will alternate between positive and negative. When dealing with GPs, establishing the common ratio is often a starting point for solving many related mathematical problems.

Quadratic Formula

The quadratic formula is a method for finding the roots of a quadratic equation. The formula is given by
\[ r = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}} \],
where
\( a \),
\( b \), and
\( c \) are the coefficients of the quadratic equation
\( ax^2 + bx + c = 0 \). The symbol
\( \pm \) indicates that there are usually two solutions or roots to a quadratic equation, known as the 'plus' and 'minus' roots.

The formula is derived from completing the square of a quadratic equation and is a critical tool in algebra. It provides a systematic way to solve any quadratic equation, even if factoring is not possible. Understanding and using the quadratic formula is essential for students as it allows for the exploration of mathematical concepts such as parabolas, and it's widely utilized in various fields of science and engineering.