Finding the Common Ratio of a Geometric Progression (G.P.)

The problem describes a Geometric Progression (G.P.) where all terms are positive. A key condition is given: each term of the G.P. is equal to the sum of the next two following terms.

Let the first term of the G.P. be \(a\) and the common ratio be \(r\). Since all terms are positive, we know that \(a > 0\). For all terms to be positive, the common ratio \(r\) must also be positive (\(r > 0\)). If \(r\) were negative, the terms would alternate in sign.

The terms of the G.P. are \(a, ar, ar^2, ar^3, \dots\).

According to the given condition, if we take any term, say the \(n\)-th term \(ar^{n-1}\), it must be equal to the sum of the next two terms, which are the \((n+1)\)-th term \(ar^n\) and the \((n+2)\)-th term \(ar^{n+1}\).

So, we can write the equation:

\(ar^{n-1} = ar^n + ar^{n+1}\)

Since \(a > 0\), we can divide both sides of the equation by \(ar^{n-1}\) (assuming \(r \neq 0\), which must be true for a G.P.).

\(\frac{ar^{n-1}}{ar^{n-1}} = \frac{ar^n}{ar^{n-1}} + \frac{ar^{n+1}}{ar^{n-1}}\)

\(1 = r^{n - (n-1)} + r^{(n+1) - (n-1)}\)

\(1 = r^1 + r^2\)

\(1 = r + r^2\)

Rearranging this equation, we get a quadratic equation in terms of \(r\):

\(r^2 + r - 1 = 0\)

We can solve this quadratic equation for \(r\) using the quadratic formula, which states that for an equation of the form \(ax^2 + bx + c = 0\), the solutions are given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In our equation \(r^2 + r - 1 = 0\), we have \(a = 1\), \(b = 1\), and \(c = -1\).

Substituting these values into the quadratic formula for \(r\):

\(r = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}\)

\(r = \frac{-1 \pm \sqrt{1 + 4}}{2}\)

\(r = \frac{-1 \pm \sqrt{5}}{2}\)

This gives us two possible values for the common ratio \(r\):

• \(r_1 = \frac{-1 + \sqrt{5}}{2}\)
• \(r_2 = \frac{-1 - \sqrt{5}}{2}\)

Now we must consider the condition that all terms of the G.P. are positive. Since the first term \(a\) is positive, the common ratio \(r\) must also be positive for all subsequent terms to be positive.

Let's evaluate the two values of \(r\):

• For \(r_1 = \frac{-1 + \sqrt{5}}{2}\): The value of \(\sqrt{5}\) is approximately 2.236. So, \(r_1 \approx \frac{-1 + 2.236}{2} = \frac{1.236}{2} \approx 0.618\). This value is positive.
• For \(r_2 = \frac{-1 - \sqrt{5}}{2}\): The value of \(\sqrt{5}\) is approximately 2.236. So, \(r_2 \approx \frac{-1 - 2.236}{2} = \frac{-3.236}{2} \approx -1.618\). This value is negative.

Since the common ratio must be positive, \(r = \frac{-1 - \sqrt{5}}{2}\) is not a valid solution for this problem.

The only valid common ratio that satisfies both the condition and the requirement that all terms are positive is \(r = \frac{-1 + \sqrt{5}}{2}\), which is often written as \(\frac{\sqrt{5} - 1}{2}\).

Revision Table: Key Concepts

Additional Information on G.P. and Related Concepts

Understanding Geometric Progressions is fundamental in sequences and series. Here are some related points:

• Sum of n terms of a G.P.: The sum of the first \(n\) terms of a G.P. is given by \(S_n = \frac{a(1 - r^n)}{1 - r}\) (when \(r \neq 1\)) or \(S_n = na\) (when \(r = 1\)).
• Sum of infinite terms of a G.P.: An infinite G.P. converges to a finite sum if and only if the absolute value of the common ratio is less than 1 (\(|r| < 1\)). The sum is given by \(S_\infty = \frac{a}{1 - r}\).
• Relationship to the Golden Ratio: The common ratio found, \(\frac{\sqrt{5}-1}{2}\), is the reciprocal of the golden ratio, \(\phi = \frac{1+\sqrt{5}}{2}\). The relationship \(r^2 + r - 1 = 0\) is closely related to the characteristic equation \(x^2 - x - 1 = 0\) associated with the Fibonacci sequence. A sequence where each term is the sum of the previous two terms (like the Fibonacci sequence) has a common ratio that approaches the golden ratio for large terms. In this problem, the condition is slightly different: each term is the sum of the *next* two terms. This leads to the reciprocal of the golden ratio.
• Applications of G.P.: G.P.s are used to model phenomena involving exponential growth or decay, such as compound interest, population growth, and radioactive decay.

The problem elegantly connects the definition of a G.P. with solving a quadratic equation and interpreting the result based on the properties of the sequence.