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Geometric Sequences: Finding the nth Term

Formula

The formula for finding the nth term in a geometric sequence is:

$a_n = a_1 \times r^{(n-1)}$

• $a_n$: The nth term of the sequence.
• $a_1$: The first term of the sequence.
• $r$: The common ratio.
• $n$: The term number.

Steps for Finding the nth Term:

1. Find the Common Ratio (r):

   • Divide any term in the geometric sequence by the term preceding it: $r = \frac{a_{n}}{a_{n-1}}$

2. Substitute Values into the Formula:

   • Use the values you’ve found to calculate the desired term using the formula $a_n = a_1 \times r^{(n-1)}$.

Examples:

• Example 1:

  • First term $a_1$ = 2
  • Common ratio $r$ = 3
  • Find the 5th term: $a_5 = 2 \times 3^{(5-1)} = 2 \times 3^4 = 2 \times 81 = 162$

• Example 2:

  • First term $a_1$ = 64
  • Common ratio $r$ = \frac{1}{2}
  • Find the 7th term: $a_7 = 64 \times \left(\frac{1}{2}\right)^{(7-1)} = 64 \times \left(\frac{1}{2}\right)^6 = 64 \times \frac{1}{64} = 1$

Additional Questions to Explore:

1. How do you determine the common ratio if you're given two non-consecutive terms in a geometric sequence?
2. What happens to the nth term if the common ratio $r$ is less than 1?
3. Can a geometric sequence have a common ratio of 0? What would be the implications?
4. How do you find the sum of the first n terms of a geometric sequence?
5. How does the geometric sequence formula change if the sequence is decreasing?

Tip: Always ensure that the common ratio is consistent throughout the sequence by checking multiple terms. This will help avoid errors when applying the formula.