How do you find the 7th term of a geometric sequence given that the first term is 4096 and the fourth term is 64?

To find the 7th term of a geometric sequence where the first term (a₁) is 4096 and the fourth term (a₄) is 64, we can use the formula for the n-th term of a geometric sequence:

• a_n = a₁ * r^(n-1)

Here, a₁ denotes the first term, r represents the common ratio, and n is the term number. We know:

• a₁ = 4096
• a₄ = 64

Using the formula for a₄, we can write:

• a₄ = a₁ * r^(4-1)
• 64 = 4096 * r³

Next, we can solve for r:

• r³ = 64 / 4096
• r³ = 1 / 64
• r³ = 64^-1
• r = 64^-1/3 = 1/4

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

• a₇ = a₁ * r^(7-1)
• a₇ = 4096 * (1/4)⁶
• a₇ = 4096 * (1/4096)
• a₇ = 1

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