Final answer:
The general term (the nth term, an) of the geometric sequence {4, 1, 16 ...} can be found using the formula an = a1 * r^(n-1), where a1 is the first term and r is the common ratio. The 7th term (a7) is 1/1024, and the sum of the first 12 terms (S12) is 1365/256.
Step-by-step explanation:
(a) The general term (the nth term, an).
In a geometric sequence, the nth term (an) can be found using the formula:
an = a1 * r^(n-1)
where a1 is the first term and r is the common ratio.
In this sequence, the first term (a1) is 4 and the common ratio (r) is 1/4. Plugging these values into the formula:
an = 4 * (1/4)^(n-1)
(b) a7, the 7th term of the sequence.
Using the formula from part (a), plugging in n = 7:
a7 = 4 * (1/4)^(7-1)
= 4 * (1/4)^6
= 4 * (1/4096)
= 1/1024
(c) The sum of the first 12 terms, S12.
The sum of a geometric sequence can be found using the formula:
Sn = a1 * (1 - r^n) / (1 - r)
Plugging in the values of a1 = 4, r = 1/4, and n = 12 into the formula:
S12 = 4 * (1 - (1/4)^12) / (1 - 1/4)
= 4 * (1 - 1/4096) / (3/4)
= 4 * (4095/4096) / (3/4)
= 1365/256