How to Find a in a Geometric Sequence?

To find a specific term (represented as a_n) in a geometric sequence, you'll typically use the formula: *a_n = a₁ r^(n-1)**. Let's break down how to use this and other approaches.

Understanding the Formula

• a_n: This is the nth term in the sequence that you want to find.
• a₁: This is the first term in the geometric sequence.
• r: This is the common ratio, which is the constant value you multiply each term by to get the next term.
• n: This is the position of the term you want to find in the sequence.

Steps to Find a_n

1. Identify a₁, r, and n:

   • a₁: Determine the first term in the sequence.
   • r: Calculate the common ratio. Divide any term by its preceding term. For example, if the sequence is 2, 6, 18, 54..., then r = 6/2 = 18/6 = 3.
   • n: Determine which term you are trying to find (e.g., the 5th term, the 10th term, etc.).

2. Plug the values into the formula: Substitute the values of a₁, r, and n into the formula a_n = a₁ * r^(n-1).

3. Calculate a_n: Simplify the equation and solve for a_n. Remember to follow the order of operations (PEMDAS/BODMAS).

Example

Let's say we have the geometric sequence: 3, 6, 12, 24,... and we want to find the 7th term (a₇).

1. Identify the values:

   • a₁ = 3 (the first term)
   • r = 6/3 = 2 (the common ratio)
   • n = 7 (we want to find the 7th term)

2. Plug into the formula:

   • a₇ = 3 * 2^(7-1)

3. Calculate:

   • a₇ = 3 * 2⁶
   • a₇ = 3 * 64
   • a₇ = 192

Therefore, the 7th term in the sequence is 192.

Finding a_n When You Know Other Terms

Sometimes, you might not know a₁ directly, but you know other terms in the sequence. In this case, you can still find a_n. You'll likely need to:

1. Find the Common Ratio (r): If you know two consecutive terms, divide the later term by the earlier term. If you know two non-consecutive terms, you'll have to use a slightly more involved method, potentially involving solving for 'r' using the general formula and the information you have.

2. Find a₁ (if needed): If you need a₁ and you know a different term (e.g., a₃) and 'r', you can rearrange the formula: a₁ = a₃ / r^(3-1)

3. Use the general formula: Once you have a₁ and 'r', you can use the formula a_n = a₁ * r^(n-1) as described earlier.

In summary, finding a specific term in a geometric sequence relies on the formula a_n = a₁ * r^(n-1). By identifying the first term, common ratio, and desired term number, you can easily calculate any term in the sequence.