We need to solve this step by step:

Step 1: Represent the first three terms in a geometric progression.

Let the first term be $a$ and the common ratio be $r$. The first three terms are: $a, ar, ar^2$ The sum of these terms is given as: a + ar + ar^2 = 112 \tag{1}

Step 2: Represent the next three terms.

The next three terms are: $ar^3, ar^4, ar^5$ The sum of these terms is given as: ar^3 + ar^4 + ar^5 = 14 \tag{2}

Step 3: Factorize both equations.

From equation (1): $a(1 + r + r^2) = 112$

From equation (2): $ar^3(1 + r + r^2) = 14$

Step 4: Divide equation (2) by equation (1).

$\frac{ar^3(1 + r + r^2)}{a(1 + r + r^2)} = \frac{14}{112}$ $r^3 = \frac{1}{8}$ $r = \frac{1}{2}$

Step 5: Solve for $a$ using $r = \frac{1}{2}$.

Substitute $r = \frac{1}{2}$ into equation (1): $a \left(1 + \frac{1}{2} + \frac{1}{4}\right) = 112$ $a \left(\frac{4}{4} + \frac{2}{4} + \frac{1}{4}\right) = 112$ $a \cdot \frac{7}{4} = 112$ $a = 64$

Step 6: General formula for the sum of terms.

The sum of the first $n$ terms of a geometric progression is: $S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r < 1$ Substitute $a = 64$ and $r = \frac{1}{2}$: $S_n = 64 \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}}$ $S_n = 64 \cdot 2 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right)$ $S_n = 128 \left(1 - \left(\frac{1}{2}\right)^n\right)$

Step 7: Determine $n$ for $S_n > 125$.

$128 \left(1 - \left(\frac{1}{2}\right)^n\right) > 125$ $1 - \left(\frac{1}{2}\right)^n > \frac{125}{128}$ $\left(\frac{1}{2}\right)^n < 1 - \frac{125}{128}$ $\left(\frac{1}{2}\right)^n < \frac{3}{128}$ Take the logarithm (base 2): $n \log_2\left(\frac{1}{2}\right) < \log_2\left(\frac{3}{128}\right)$ $-n < \log_2(3) - 7$ $n > 7 - \log_2(3)$ Using $\log_2(3) \approx 1.585$: $n > 7 - 1.585$ $n > 5.415$ Since $n$ must be an integer, $n = 6$.

Final Answer:

It takes 6 terms for the sum to exceed 125.

Would you like a more detailed explanation or clarification on any step?
Here are some related questions to explore:

1. How do logarithmic properties help solve exponential equations?
2. What is the derivation of the sum formula for a geometric progression?
3. How does the common ratio influence the growth or decay of a sequence?
4. Can this result be verified using numerical methods or simulations?
5. How does the choice of base for the logarithm affect the calculation?

Tip: Always verify calculations involving logarithms with precise values or a calculator for better accuracy!