The formula a_n = a_1 * r^(n-1) is used to describe the nth term of a geometric sequence. It relates the nth term (a_n) to the first term (a_1) and the common ratio (r) of the sequence. This formula allows you to calculate any term in the sequence once you know the first term and the common ratio.
congrats on reading the definition of a_n = a_1 * r^(n-1). now let's actually learn it.
The formula a_n = a_1 * r^(n-1) is the explicit formula for a geometric sequence.
The variable 'a_n' represents the nth term of the sequence, 'a_1' represents the first term, and 'r' represents the common ratio.
The exponent '(n-1)' accounts for the fact that the nth term is reached after (n-1) multiplications by the common ratio.
This formula is useful for quickly calculating any term in the sequence without having to work through all the previous terms.
Knowing the first term and common ratio allows you to use this formula to generate the entire sequence.
Review Questions
Explain how the formula a_n = a_1 * r^(n-1) relates the nth term to the first term and common ratio in a geometric sequence.
The formula a_n = a_1 * r^(n-1) shows that the nth term of a geometric sequence (a_n) is equal to the first term (a_1) multiplied by the common ratio (r) raised to the power of (n-1). This means that each term in the sequence is found by taking the previous term and multiplying it by the common ratio. The exponent (n-1) accounts for the fact that the nth term is reached after (n-1) multiplications by the common ratio, starting from the first term.
Describe how you would use the formula a_n = a_1 * r^(n-1) to find the 10th term of a geometric sequence if you know the first term is 3 and the common ratio is 0.5.
To find the 10th term of the geometric sequence using the formula a_n = a_1 * r^(n-1), we would plug in the known values:
* a_1 = 3 (the first term)
* r = 0.5 (the common ratio)
* n = 10 (the term we want to find)
Plugging these values into the formula, we get:
a_10 = 3 * (0.5)^(10-1)
a_10 = 3 * (0.5)^9
a_10 = 3 * 0.001953125
a_10 = 0.005859375
Therefore, the 10th term of the sequence is 0.005859375.
Analyze how the formula a_n = a_1 * r^(n-1) can be used to determine the behavior of a geometric sequence as the number of terms increases, particularly in regards to sequences that have a common ratio greater than 1 or less than 1.
The formula a_n = a_1 * r^(n-1) reveals important information about the behavior of a geometric sequence as the number of terms increases:
* If the common ratio r > 1, then as n increases, r^(n-1) will become larger, causing the terms to grow exponentially larger. This results in a sequence that diverges to positive infinity.
* If the common ratio r < 1, then as n increases, r^(n-1) will become smaller, causing the terms to approach 0. This results in a sequence that converges to 0.
* The formula shows that the growth or decay of the sequence is determined by the value of the common ratio r. Sequences with r > 1 will grow without bound, while sequences with r < 1 will approach 0 as more terms are added.