Final answer:
The first term of a geometric sequence with a sixth term of 128 and a common ratio of 2 is 4. This is found by dividing 128 by 2 raised to the power of 5 (32), which is the growth factor from the first to the sixth term.
Step-by-step explanation:
If the sixth term of a geometric sequence is 128 and the common ratio is 2, we can find the first term using the formula for the nth term of a geometric sequence, which is an = a1 × r(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number. Since we know the sixth term (a6 = 128) and the common ratio (r = 2), we can plug these values into the formula and solve for the first term (a1).
Using the formula, we get 128 = a1 × 2(6-1) = a1 × 25. Since 25 equals 32, we can divide both sides of the equation by 32 to isolate a1. This gives us a1 = 128 / 32 = 4. Therefore, the first term of the sequence is 4.