The image you provided shows a
Geometric Sequence Calculator interface. The formula for finding the \( n \)-th term of a geometric sequence is given as:
\[
a_n = a_1 \cdot r^{n-1}
\]
Explanation of the Formula:
1.
\( a_n \): This represents the \( n \)-th term of the geometric sequence.
2.
\( a_1 \): This is the first term of the geometric sequence.
3.
\( r \): This is the common ratio between consecutive terms in the sequence.
4.
\( n \): This is the position of the term in the sequence.
The formula \( a_n = a_1 \cdot r^{n-1} \) allows you to calculate any term in the sequence if you know the first term (\( a_1 \)), the common ratio (\( r \)), and the position (\( n \)) of the term you want to find.
Steps to Solve a Problem Using This Formula:
1. Identify the values of \( a_1 \), \( r \), and \( n \).
2. Substitute these values into the formula \( a_n = a_1 \cdot r^{n-1} \).
3. Simplify the expression to find \( a_n \).
Example Problem:
Suppose you are given:
- The first term \( a_1 = 2 \),
- The common ratio \( r = 3 \),
- You want to find the 5th term (\( n = 5 \)).
#### Solution:
1. Use the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
2. Substitute the given values:
\[
a_5 = 2 \cdot 3^{5-1}
\]
3. Simplify the exponent:
\[
a_5 = 2 \cdot 3^4
\]
4. Calculate \( 3^4 \):
\[
3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81
\]
5. Multiply by the first term:
\[
a_5 = 2 \cdot 81 = 162
\]
Thus, the 5th term of the geometric sequence is \( \boxed{162} \).
General Approach:
- Always ensure you have the correct values for \( a_1 \), \( r \), and \( n \).
- Substitute these values into the formula.
- Simplify step by step to find the desired term.
If you have specific values or a particular problem you'd like to solve, feel free to provide them, and I can walk you through the solution!
Parent Tip: Review the logic above to help your child master the concept of geometric sequences.
