What is the 7th term of the geometric sequence where a1 = 4096 and a4 = 64?

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 is obtained by multiplying the previous term by a constant ratio, which we’ll denote as r.

We start with the information given:

• First term, a₁ = 4096
• Fourth term, a₄ = 64

The formula for the nth term of a geometric sequence is:

a_n = a₁ imes r^n-1

Using this formula for the fourth term:

a₄ = a₁ imes r^4-1 = a₁ imes r³

Substituting the values we know:

64 = 4096 imes r³

To isolate r³, we divide both sides by 4096:

r³ = 64 / 4096

r³ = 1 / 64

Now, to solve for r, we take the cube root of both sides:

r = (1 / 64)^1/3

r = 1 / 4

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

a₇ = a₁ imes r^7-1 = 4096 imes r⁶

Calculating r⁶:

r⁶ = (1 / 4)⁶ = 1 / 4096

Now we can substitute this back into our equation for a₇:

a₇ = 4096 imes (1 / 4096) = 1

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