Key Concepts and Formulas

• Geometric Progression (G.P.): A sequence 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 $a, ar, ar^2, \dots$.
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• Greatest Integer Function (Floor Function): $[x]$ denotes the greatest integer less than or equal to $x$. For a non-integer $x$, $[x] + [-x] = -1$.

Step-by-Step Solution

Step 1: Define the terms of the G.P. and set up the triangle inequality. Let the three successive terms of the G.P. be $a$, $ar$, and $ar^2$. Since these are lengths of sides of a triangle, $a > 0$. We are given that the common ratio $r > 1$. This implies that the terms are ordered: $a < ar < ar^2$. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides must be greater than the length of the third side. The three inequalities are:

1. $a + ar > ar^2$
2. $a + ar^2 > ar$
3. $ar + ar^2 > a$

Since $r > 1$ and $a > 0$, $ar^2$ is the longest side. Inequality (2) ($a + ar^2 > ar$) and inequality (3) ($ar + ar^2 > a$) are always satisfied because $ar^2 > ar$, $ar^2 > a$, and $ar > a$. Therefore, we only need to consider the inequality where the sum of the two shorter sides is greater than the longest side: $a + ar > ar^2$

Step 2: Derive the range for the common ratio $r$. Divide the inequality $a + ar > ar^2$ by $a$ (since $a > 0$, the inequality direction remains unchanged): $1 + r > r^2$ Rearrange the terms to form a quadratic inequality: $r^2 - r - 1 < 0$ To find the values of $r$ that satisfy this inequality, we first find the roots of the quadratic equation $r^2 - r - 1 = 0$. Using the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $r = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{1+4}}{2} = \frac{1 \pm \sqrt{5}}{2}$ The roots are $r_1 = \frac{1 - \sqrt{5}}{2}$ and $r_2 = \frac{1 + \sqrt{5}}{2}$. Since the quadratic $r^2 - r - 1$ has a positive leading coefficient, the inequality $r^2 - r - 1 < 0$ is satisfied for $r$ values between the roots: $\frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2}$ We are given that $r > 1$. We also know that $\sqrt{5} \approx 2.236$. So, $\frac{1 - \sqrt{5}}{2} \approx \frac{1 - 2.236}{2} \approx -0.618$, and $\frac{1 + \sqrt{5}}{2} \approx \frac{1 + 2.236}{2} \approx 1.618$. Thus, the inequality becomes $-0.618 < r < 1.618$. Combining this with the given condition $r > 1$, the valid range for $r$ is: $1 < r < \frac{1 + \sqrt{5}}{2}$

Step 3: Determine the values of $[r]$ and $[-r]$. From Step 2, we have $1 < r < \frac{1 + \sqrt{5}}{2} \approx 1.618$. The greatest integer less than or equal to any number in this interval is $1$. Therefore, $[r] = 1$.

Now, let's find the range for $-r$. Multiplying the inequality $1 < r < \frac{1 + \sqrt{5}}{2}$ by $-1$ and reversing the inequality signs: $-\frac{1 + \sqrt{5}}{2} < -r < -1$ Approximately, this is $-1.618 < -r < -1$. The greatest integer less than or equal to any number in the interval $(-1.618, -1)$ is $-2$. Therefore, $[-r] = -2$.

Step 4: Calculate the value of the expression $3[r] + [-r]$. Substitute the values found in Step 3 into the given expression: $3[r] + [-r] = 3(1) + (-2)$ $= 3 - 2$ $= 1$

Common Mistakes & Tips

• Triangle Inequality Simplification: Always verify which inequalities are redundant given the ordering of sides. For a G.P. with $r>1$, only the sum of the two smaller sides needs to be greater than the largest side.
• Floor Function Properties: Be careful with the floor function of negative numbers. For a non-integer $x$, $[x] + [-x] = -1$. In this problem, $r$ is not an integer, so $[r] + [-r] = -1$, which is consistent with our findings: $1 + (-2) = -1$.
• Combining Inequalities: Ensure that all given conditions (like $r>1$) are combined with the derived inequalities to find the final valid range for $r$.

Summary

The problem requires us to use the triangle inequality theorem for three successive terms of a geometric progression with common ratio $r > 1$. This leads to the condition $1 < r < \frac{1+\sqrt{5}}{2}$. We then applied the definition of the greatest integer function to find $[r]=1$ and $[-r]=-2$. Substituting these values into the expression $3[r] + [-r]$ gives the final answer.

The final answer is $\boxed{1}$.