Find the Nth Term Using Explicit Formula - Algebra-1

1. Fundamental Concepts

Explicit Formula for the nth Term: In a geometric sequence, the explicit formula is used to directly calculate the nth term () without needing to find all preceding terms. It is expressed as: where:
- = the nth term of the sequence,
- = the first term,
- r = the common ratio,
- n = the term number (positive integer).

Independence of the Formula: The explicit formula allows calculating any term () using only , r, and n, making it efficient for finding distant terms (e.g., the 20th term).
Role of Each Component:
- is the starting value of the sequence.
- r determines the growth or decay of the sequence (e.g., leads to growth, leads to decay).
- The exponent accounts for the number of multiplications by r needed to reach the nth term from (e.g., the 3rd term requires multiplying by r twice: ).
Applicability: The formula only works for geometric sequences (where the common ratio r is constant).

Question: A geometric sequence has a first term and a common ratio . Find the 7th term.
Solution:
- Given , , .
- Use the formula: .
- Simplify: , so .
Conclusion: The 7th term is .

Question: Find the 6th term of the geometric sequence using the explicit formula.
Solution:
- Identify , , .
- Apply the formula: .
- Calculate: , so .
Conclusion: The 6th term is 486.

Question: A geometric sequence has a 3rd term and a common ratio . Find the 10th term.
Solution:
- Step 1: Find using the 3rd term. From , substitute and :.
- Step 2: Calculate the 10th term with , , :.
- Simplify: , so .
Conclusion: The 10th term is 39366.

Identify Known Values: First, determine , r, and n from the problem. If any of these are missing (e.g., or r), use given terms to solve for them first.
Substitute into the Formula: Plug the known values into . Pay attention to the exponent (not n) to avoid errors.
Simplify Exponents Carefully: For negative r, remember that even exponents result in positive values, and odd exponents result in negative values (e.g., , ).
Verify with Smaller Terms: If unsure, check the formula with a known term (e.g., calculate the 2nd or 3rd term using the formula and compare it to the given sequence).