The series given is a geometric progression: $75 + 300 + 1200 + \dots + 4915200$

In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio $r$. To identify how many terms are in the series, we need to:

Step 1: Identify the first term and the common ratio

• The first term $a_1$ is $75$.
• To find the common ratio $r$, divide the second term by the first term: $r = \frac{300}{75} = 4$ Thus, the common ratio is $r = 4$.

Step 2: General term of the geometric series

The $n$-th term of a geometric series is given by the formula: $a_n = a_1 \cdot r^{n-1}$ Where:

• $a_1 = 75$ (the first term),
• $r = 4$ (the common ratio).

We are given that the last term of the series is $a_n = 4915200$. Now, we can set up the equation to find $n$: $4915200 = 75 \cdot 4^{n-1}$

Step 3: Solve for $n$

Divide both sides by 75: $\frac{4915200}{75} = 4^{n-1}$ $65536 = 4^{n-1}$ Now, express $65536$ as a power of 4: $65536 = 4^8$ Thus, we have: $4^{n-1} = 4^8$ Equating the exponents: $n - 1 = 8 \implies n = 9$

Final Answer:

There are $\boxed{9}$ terms in the geometric series.

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Do you have any questions or would you like more details?
Here are 5 related questions for further exploration:

1. How do you derive the sum of the first $n$ terms of a geometric series?
2. What is the sum of this particular geometric series?
3. How do geometric progressions apply in compound interest problems?
4. How would the number of terms change if the last term were $9830400$?
5. Can you explain how geometric sequences differ from arithmetic sequences?

Tip: In any geometric progression, the common ratio helps in quickly identifying the growth pattern and can be useful in real-world scenarios like population growth and investments!