The given sequence is: 3, 12, 48, 192.
To find the equation for the nth term, we first examine how the sequence is generated. We can observe the following pattern:
- The first term (n=1) is 3.
- The second term (n=2) is 12, which is 3 x 4.
- The third term (n=3) is 48, which is 12 x 4.
- The fourth term (n=4) is 192, which is 48 x 4.
From this, we can see that each term can be expressed as:
- an = an-1 x 4
Starting with an initial value (a1 = 3), we can express the terms of the sequence recursively:
- a2 = 3 x 4 = 12
- a3 = 12 x 4 = 48
- a4 = 48 x 4 = 192
To derive a formula for the nth term explicitly, we can write:
an = a1 x 4(n-1) where a1 = 3.
Thus, substituting the value of a1, we get:
an = 3 x 4(n-1)
So, the equation for the nth term of the sequence is:
an = 3 x 4(n-1)
This formula indicates a geometric sequence where the first term is 3 and the common ratio is 4.