Calculating the nth term of a geometric sequence using the explicit formula with an example

Explicit Formula for Geometric Sequence

The explicit formula for a geometric sequence is given by:

an = a1 * r^(n-1)

where:
– an represents the nth term in the sequence
– a1 represents the first term in the sequence
– r represents the common ratio between consecutive terms
– n represents the position of the term in the sequence

Let’s break down the formula with an example:

Consider the geometric sequence: 2, 4, 8, 16, 32,

The explicit formula for a geometric sequence is given by:

an = a1 * r^(n-1)

where:
– an represents the nth term in the sequence
– a1 represents the first term in the sequence
– r represents the common ratio between consecutive terms
– n represents the position of the term in the sequence

Let’s break down the formula with an example:

Consider the geometric sequence: 2, 4, 8, 16, 32, …

To find the explicit formula, we need to determine the values of a1 and r.

Looking at the first two terms, we can see that the first term, a1, is 2, and the second term is 4. To find the common ratio, we divide any term by its preceding term:

common ratio (r) = term 2 / term 1 = 4 / 2 = 2

Now that we have a1 = 2 and r = 2, we can substitute these values into the explicit formula:

an = 2 * 2^(n-1)

This formula will give us any term in the sequence when we specify its position, n.

For example, let’s find the 6th term (n = 6):

a6 = 2 * 2^(6-1)
a6 = 2 * 2^5
a6 = 2 * 32
a6 = 64

Therefore, the 6th term in the given geometric sequence is 64.

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