Write an explicit formula for a, start subscript, n, end subscripta 
n
​	
 , the n, start superscript, th, end superscriptn 
th
  term of the sequence 7, comma, 35, comma, 175, comma, point, point, .7,35,175,....


The given sequence is:

\[ 7, 35, 175, \dots \]

To identify the pattern, we can look at the ratio between consecutive terms:

\[
\frac{35}{7} = 5 \quad \text{and} \quad \frac{175}{35} = 5
\]

Since the ratio between consecutive terms is constant, this is a **geometric sequence** with a common ratio \( r = 5 \) and the first term \( a_1 = 7 \).

The general formula for the \( n \)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

Substitute \( a_1 = 7 \) and \( r = 5 \):

\[
a_n = 7 \cdot 5^{n-1}
\]

Thus, the explicit formula for the \( n \)-th term of the sequence is:

\[
a_n = 7 \cdot 5^{n-1}
\]

Would you like more details or have any questions?

Here are 5 questions that expand on this concept:
1. How do you derive the formula for the \( n \)-th term of a geometric sequence?
2. What is the 6th term in the sequence using the formula \( a_n = 7 \cdot 5^{n-1} \)?
3. What happens if the common ratio \( r \) is less than 1 in a geometric sequence?
4. Can you determine the sum of the first 5 terms of the given sequence?
5. How does a geometric sequence differ from an arithmetic sequence?

**Tip:** In a geometric sequence, if the common ratio is greater than 1, the terms grow exponentially.

Write an explicit formula for aₙ, the nᵗʰ term of the sequence 7, 35, 175, ....

Discover how to derive the explicit formula for the nᵗʰ term of a geometric sequence with the terms 7, 35, 175, and more. Learn about geometric sequences, common ratios, and exponential growth. Suitable for high school students in Grades 9-12.

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