Key Concepts and Formulas

• Geometric Progression (GP): 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 ($r$). The terms are of the form $a, ar, ar^2, ar^3, \dots$.
• Quadratic Formula: For a quadratic equation of the form $Ax^2 + Bx + C = 0$, the solutions for $x$ are given by $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$.

Step-by-Step Solution

• Step 1: Define the terms of the GP and set up the given condition. Let the first term of the geometric progression be $a$ and the common ratio be $r$. The terms of the GP are $T_1 = a$, $T_2 = ar$, $T_3 = ar^2$, and so on. The problem states that each term equals the sum of the next two terms. Applying this to the first term ($T_1$): $T_1 = T_2 + T_3$

• Step 2: Substitute the general terms of the GP into the equation. Substitute $T_1 = a$, $T_2 = ar$, and $T_3 = ar^2$ into the equation from Step 1: $a = ar + ar^2$

• Step 3: Simplify the equation by dividing by the first term $a$. The problem states that the GP consists of positive terms, which implies that the first term $a$ must be positive ($a > 0$). Since $a$ is non-zero, we can divide the entire equation by $a$: $\frac{a}{a} = \frac{ar}{a} + \frac{ar^2}{a}$ $1 = r + r^2$

• Step 4: Rearrange the equation into a standard quadratic form. To solve for the common ratio $r$, rearrange the equation into the standard quadratic form $Ar^2 + Br + C = 0$: $r^2 + r - 1 = 0$

• Step 5: Solve the quadratic equation for $r$ using the quadratic formula. Using the quadratic formula with $A=1$, $B=1$, and $C=-1$: $r = \frac{-1 \pm \sqrt{(1)^2 - 4(1)(-1)}}{2(1)}$ $r = \frac{-1 \pm \sqrt{1 + 4}}{2}$ $r = \frac{-1 \pm \sqrt{5}}{2}$ This gives two possible values for $r$: $r_1 = \frac{-1 + \sqrt{5}}{2}$ and $r_2 = \frac{-1 - \sqrt{5}}{2}$.

• Step 6: Determine the correct value of $r$ based on the problem's constraint. The problem states that the geometric progression consists of positive terms. If the first term $a$ is positive, then for all terms to be positive, the common ratio $r$ must also be positive. Let's examine the two possible values for $r$:

  • $r_1 = \frac{-1 + \sqrt{5}}{2}$: Since $\sqrt{5} \approx 2.236$, this value is approximately $\frac{-1 + 2.236}{2} = \frac{1.236}{2} \approx 0.618$, which is positive.
  • $r_2 = \frac{-1 - \sqrt{5}}{2}$: This value is approximately $\frac{-1 - 2.236}{2} = \frac{-3.236}{2} \approx -1.618$, which is negative. Therefore, to ensure all terms are positive, we must choose the positive value of $r$. $r = \frac{\sqrt{5} - 1}{2}$

Common Mistakes & Tips

• Ignoring the "positive terms" condition: This is a crucial constraint that helps in selecting the correct root of the quadratic equation. Without it, both roots would be mathematically valid but only one fits the problem's context.
• Algebraic errors in solving the quadratic equation: Carefully apply the quadratic formula and perform the arithmetic accurately.
• Dividing by zero: Ensure that the term you are dividing by (in this case, $a$) is non-zero, which is guaranteed by the condition of positive terms.

Summary

We translated the problem statement about a geometric progression where each term equals the sum of the next two into an algebraic equation involving the first term $a$ and the common ratio $r$. This led to a quadratic equation in $r$. Solving this quadratic equation yielded two possible values for $r$. The condition that all terms in the progression are positive allowed us to select the unique positive value for the common ratio, $r = \frac{\sqrt{5} - 1}{2}$.

The final answer is $\boxed{{\frac{\sqrt 5 - 1}{2}}}$.