Write the explicit formula for the geometric sequence.
4, 3, 2 1/4, 1 11/16, . . .
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html
4 * 3/4 = 3
3 * 3/4 = 9/4 = 2 1/4
9/4*3/4 = 27/16 = 1 11/16 remarkable, we might decide r = 3/4
and the first term a = 4
Thank you
To find the explicit formula for the geometric sequence 4, 3, 2 1/4, 1 11/16, and so on, we need to determine the common ratio between consecutive terms.
To do this, we can divide any term by its previous term. Let's take the second term divided by the first term:
3 ÷ 4 = 3/4
We can simplify 3/4 as a decimal:
3/4 = 0.75
Now, let's take the third term divided by the second term:
2 1/4 ÷ 3 = 9/4 ÷ 3 = 9/12 = 3/4
Similarly, let's divide the fourth term by the third term:
1 11/16 ÷ 2 1/4 = 19/16 ÷ 9/4 = 19/16 × 4/9 = 19/36
Based on these calculations, we can see that the common ratio between consecutive terms in this geometric sequence is 3/4.
To find the explicit formula for a geometric sequence with a given first term (a) and common ratio (r), we use the formula:
nth term = a × r^(n-1), where nth term represents the term we are interested in and n is the position of that term in the sequence (starting from 1).
So, for the given sequence, the explicit formula is:
nth term = 4 × (3/4)^(n-1)
To find the explicit formula for a geometric sequence, we need to determine the common ratio, denoted by "r," which is the ratio between any two consecutive terms.
In this case, let's find the ratio between the second and first terms:
r = 3/4
Now, we can use the general formula for a geometric sequence:
a_n = a_1 * r^(n-1)
Here, "a_n" represents the nth term of the sequence, "a_1" is the first term, "r" is the common ratio, and "n" is the position of the term we want to find.
To apply this formula to the given sequence:
a_n = 4 * (3/4)^(n-1)
Therefore, the explicit formula for the given geometric sequence is a_n = 4 * (3/4)^(n-1).