Write an example of a geometric sequence with a t(1) = 7, give at least the first 4 terms. Explain how you know that it is Geometric.
example 1
pick a value of r, let's pick r = 3
sequence: 7, 21, 63, 189, 567, ...
example 2
pick a value of r, let's pick r = -2
sequence: 7, -14 , 28, ..... (find 2 more)
example 3
.....
example 4
....
We know they are geometric, because we used the definition
Arithmetic sequences
To create a geometric sequence, we need to multiply each term by a constant ratio to get the next term. Let's use a common ratio of 2 for this example.
The formula for a geometric sequence is given as:
t(n) = t(1) * r^(n-1)
Given t(1) = 7, and a common ratio r = 2, we can calculate the first few terms as follows:
t(1) = 7
t(2) = t(1) * r^1 = 7 * 2^1 = 7 * 2 = 14
t(3) = t(1) * r^2 = 7 * 2^2 = 7 * 4 = 28
t(4) = t(1) * r^3 = 7 * 2^3 = 7 * 8 = 56
So, the first four terms of the geometric sequence with t(1) = 7, and a common ratio of 2 are: 7, 14, 28, 56.
We know this is a geometric sequence because each term is obtained by multiplying the previous term by the constant ratio of 2.