Let's define a sequence and find its first 10 terms along with the equation.

Geometric Sequence Example

Consider a geometric sequence where:

• The first term ($a_1$) is 2.
• The common ratio ($r$) is 3.

Equation for the $n$th Term:

The equation for the $n$th term ($a_n$) of a geometric sequence is given by: $a_n = a_1 \cdot r^{n-1}$

Calculating the First 10 Terms:

• $a_1 = 2 \cdot 3^{1-1} = 2$
• $a_2 = 2 \cdot 3^{2-1} = 6$
• $a_3 = 2 \cdot 3^{3-1} = 18$
• $a_4 = 2 \cdot 3^{4-1} = 54$
• $a_5 = 2 \cdot 3^{5-1} = 162$
• $a_6 = 2 \cdot 3^{6-1} = 486$
• $a_7 = 2 \cdot 3^{7-1} = 1458$
• $a_8 = 2 \cdot 3^{8-1} = 4374$
• $a_9 = 2 \cdot 3^{9-1} = 13122$
• $a_{10} = 2 \cdot 3^{10-1} = 39366$

First 10 Terms of the Sequence:

$2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366$

Would you like further details or have any questions?

Related Questions:

1. How do you derive the equation for a geometric sequence?
2. What if the common ratio $r$ is a fraction or a negative number?
3. How does a geometric sequence differ from an arithmetic sequence?
4. Can you find the sum of the first 10 terms of this sequence?
5. How would you find the 50th term of this sequence?

Tip:

When solving sequence problems, always verify whether the sequence is arithmetic or geometric by checking if there is a constant difference or ratio between terms.