Identify the 12th term of a geometric sequence where a1 is 8 and a6 is 8192

To find the 12th term of the geometric sequence, we first need to understand the formula for the nth term of a geometric sequence, which is given by:

a_n = a₁ * r^(n-1)

Here, a₁ is the first term, r is the common ratio, and n is the term number. Given that a₁ = 8 and a₆ = 8192, we can use this information to find r.

First, let’s use the formula to express a₆:

a₆ = a₁ * r^(6-1) = 8 * r⁵

Setting this equal to 8192 gives:

8 * r⁵ = 8192

To solve for r⁵, divide both sides by 8:

r⁵ = 8192 / 8 = 1024

Next, we need to find r. We can rewrite 1024 as a power of 2:

1024 = 2¹⁰

Now, since r⁵ = 2¹⁰, we can express r as:

r = (2¹⁰)^(1/5) = 2² = 4

Now that we have r = 4, we can find the 12th term:

a₁₂ = a₁ * r^(12-1) = 8 * 4¹¹

Calculating 4¹¹:

4¹¹ = (2²)¹¹ = 2²²

Now express a₁₂:

a₁₂ = 8 * 2²²

Since 8 = 2³:

a₁₂ = 2³ * 2²² = 2²⁵

Finally, simplifying gives:

a₁₂ = 33554432

Therefore, the 12th term of the geometric sequence is 33,554,432.