How do I write equations for geometric sequences?
example question:
1, 3/2, 9/4, 27/8
an= a(n-1) * r^n where r is the value of dividing one term by the term before.
For instance, in the above, ao= 1,
r= a3/a2= (27/8)/(9/4)= 3/2
a1= 1*3/2
a3= 1*(3/2)^3= 27/8
Thank you, Mr. Pursley
To write equations for geometric sequences, you need to know the initial term and the common ratio between consecutive terms.
Let's use the given example: 1, 3/2, 9/4, 27/8.
To start, we have the initial term a₀ = 1.
Next, to find the common ratio (r), we divide any term by the term before it. For example, to find r for the second term, we divide the second term (3/2) by the first term (1):
r = (3/2) / 1 = 3/2
Now that we have the initial term (a₀) and the common ratio (r), we can write the equation for the nth term (aₙ) in a geometric sequence as follows:
aₙ = aₙ₋₁ * rⁿ
In this equation, aₙ represents the nth term, aₙ₋₁ represents the (n-1)th term, and rⁿ represents r raised to the power of n.
Using the values we found earlier, the equation for this geometric sequence becomes:
aₙ = 1 * (3/2)ⁿ
For example, if we want to find the third term (a₃), we can plug in n = 3 into the equation:
a₃ = 1 * (3/2)³ = 27/8
So, the third term of this sequence is 27/8.