What is the 7th term of the geometric sequence where a1 is 1024 and a4 is 16?

To find the 7th term of the geometric sequence, we first need to determine the common ratio of the sequence.

In a geometric sequence, each term can be represented as follows:

• a_n = a₁ * r^(n-1)

Here, a₁ = 1024 and a₄ = 16.

We can express a₄ in terms of a₁ and the common ratio r:

a₄ = a₁ * r^(4-1) = a₁ * r³

Substituting the known values:

16 = 1024 * r³

Now, to isolate r³, we divide both sides by 1024:

r³ = 16 / 1024

r³ = 1 / 64

Now, take the cube root of both sides to find r:

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

Now that we have r, we can find the 7th term, a₇:

a₇ = a₁ * r^(7-1) = 1024 * r⁶

Substituting the value of r:

a₇ = 1024 * (1 / 4)⁶

Calculating (1/4)⁶:

(1/4)⁶ = 1 / 4096

So, a₇ = 1024 * (1 / 4096) = 1024 / 4096 = 1 / 4

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