can someone please make me create a geometric sequence up to 5 terms? so that I can find the explicit formula for the nth term
pick any initial term a and common ratio r.
The nth term is always an = a*r^(n-1)
Of course! I can guide you on how to create a geometric sequence and find the explicit formula for the nth term.
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant value called the common ratio. To create a geometric sequence, follow these steps:
1. Choose a starting term, denoted as a.
2. Determine the common ratio, denoted as r.
For example, let's create a geometric sequence with a starting term of 2 and a common ratio of 3.
Term 1: a = 2
Term 2: multiply the previous term (2) by the common ratio (3): 2 * 3 = 6
Term 3: multiply the previous term (6) by the common ratio (3): 6 * 3 = 18
Term 4: multiply the previous term (18) by the common ratio (3): 18 * 3 = 54
Term 5: multiply the previous term (54) by the common ratio (3): 54 * 3 = 162
Now that we have the terms of the geometric sequence (2, 6, 18, 54, 162), let's find the explicit formula for the nth term.
The explicit formula for a geometric sequence is given by the equation:
an = a * (r^(n-1))
where an represents the nth term, a is the first term, r is the common ratio.
Using our example values, we can substitute them into the formula:
a = 2 (the first term)
r = 3 (the common ratio)
n = 1, 2, 3, 4, 5 (the terms)
To find the explicit formula for each term, substitute the appropriate values into the formula:
Term 1: a1 = 2 * (3^(1-1)) = 2 * 3^0 = 2 * 1 = 2
Term 2: a2 = 2 * (3^(2-1)) = 2 * 3^1 = 2 * 3 = 6
Term 3: a3 = 2 * (3^(3-1)) = 2 * 3^2 = 2 * 9 = 18
Term 4: a4 = 2 * (3^(4-1)) = 2 * 3^3 = 2 * 27 = 54
Term 5: a5 = 2 * (3^(5-1)) = 2 * 3^4 = 2 * 81 = 162
So, the explicit formula for the nth term in our geometric sequence is an = 2 * (3^(n-1)).