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The problem provided deals with a geometric series where each jump is \( \frac{2}{3} \) of the previous jump. 

Given:
- The first jump is 27 cm.
- The common ratio \( r \) is \( \frac{2}{3} \).

We need to find the total distance covered after 5 leaps.

### Formula for the sum of the first \( n \) terms of a geometric series:
\[
S_n = a \cdot \frac{1 - r^n}{1 - r}
\]
where:
- \( S_n \) is the sum after \( n \) jumps,
- \( a \) is the first term (27 cm),
- \( r \) is the common ratio \( \left( \frac{2}{3} \right) \),
- \( n \) is the number of terms (in this case, 5).

Let's calculate the total distance after 5 leaps.The total distance the frog has covered after 5 leaps is approximately 70.33 cm.

Would you like more details or further explanations on this? 

Here are 5 related questions to deepen understanding:
1. How would the total distance change if the frog jumped 6 times instead of 5?
2. What happens to the total distance if the common ratio is changed to \( \frac{1}{2} \)?
3. How is the sum of an infinite geometric series different from the sum of a finite geometric series?
4. What is the distance covered after the third leap alone, not cumulatively?
5. How does the first jump influence the entire geometric series?

**Tip:** In geometric series, if the common ratio \( |r| < 1 \), the sum of an infinite number of terms approaches a limit.

A frog leaps ⅔ of the previous jump. If the frog's first leap is 27 cm, find the distance the frog has covered after 5 leaps.

Learn how to solve a geometric series problem where a frog leaps ⅔ of the previous jump. With the first leap of 27 cm, calculate the total distance covered after 5 leaps. Suitable for students in Grades 8-10.

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