What is the 7th term of the geometric sequence where the first term is 1024 and the fourth term is 16?

To find the 7th term of the geometric sequence where the first term (a₁) is 1024 and the fourth term (a₄) is 16, we can follow these steps:

A geometric sequence is characterized by a constant ratio r between successive terms. The nth term of a geometric sequence can be expressed as:

a_n = a₁ imes r^(n-1)

Given:

• a₁ = 1024
• a₄ = 16

Using the formula for the fourth term:

a₄ = a₁ imes r^(4-1) = 1024 imes r³

Substituting the known value:

16 = 1024 imes r³

To solve for r, we first isolate r³:

r³ = 16 / 1024

Calculating the right side:

r³ = 1 / 64

Now, taking the cube root of both sides:

r = (1 / 64)^(1/3) = 1 / 4

Now that we have the common ratio r, we can find the 7th term:

a₇ = a₁ imes r^(7-1) = 1024 imes r⁶

Substituting r back in:

a₇ = 1024 imes (1 / 4)⁶

Calculating (1 / 4)⁶:

(1 / 4)⁶ = 1 / 4096

Now we can find a₇:

a₇ = 1024 imes (1 / 4096)

a₇ = 1024 / 4096

a₇ = 1 / 4

Thus, the 7th term of the geometric sequence is 1/4.