Question
Find the sum of the eight term of 12,6,3
Answers
GPT 3.5
To find the sum of the eighth term, we need to determine the pattern in the given sequence.
Starting with 12, the common ratio between each term is 1/2.
So, the next term after 12 is (12)(1/2) = 6.
The next term after 6 is (6)(1/2) = 3.
The next term after 3 is (3)(1/2) = 3/2.
We can observe that the sequence alternates between 3 and 3/2.
To find the eighth term, we start at 12 and repeatedly divide by 2 a total of 3 times, since we are looking for the eighth term. This gives us:
(12)(1/2)(1/2)(1/2) = 12/8 = 3/2
So, the eighth term is 3/2.
To find the sum of the eighth term, we add up all the terms in the given sequence. The sum of n terms in a geometric sequence can be found using the formula:
Sn = a(1 - r^n) / (1 - r)
Where Sn is the sum of the n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 12, r = 1/2, and n = 8.
Plugging in these values into the formula, we get:
S8 = 12(1 - (1/2)^8) / (1 - 1/2)
Simplifying, we have:
S8 = (12 - 3/256) / (1/2)
Multiplying by the reciprocal, we get:
S8 = (12 - 3/256)(2/1)
S8 = (24 - 3/128)
Now we can convert 24 to a fraction with the same denominator:
S8 = (3072/128 - 3/128)
S8 = (3069/128)
Thus, the sum of the eighth term is 3069/128.
Starting with 12, the common ratio between each term is 1/2.
So, the next term after 12 is (12)(1/2) = 6.
The next term after 6 is (6)(1/2) = 3.
The next term after 3 is (3)(1/2) = 3/2.
We can observe that the sequence alternates between 3 and 3/2.
To find the eighth term, we start at 12 and repeatedly divide by 2 a total of 3 times, since we are looking for the eighth term. This gives us:
(12)(1/2)(1/2)(1/2) = 12/8 = 3/2
So, the eighth term is 3/2.
To find the sum of the eighth term, we add up all the terms in the given sequence. The sum of n terms in a geometric sequence can be found using the formula:
Sn = a(1 - r^n) / (1 - r)
Where Sn is the sum of the n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 12, r = 1/2, and n = 8.
Plugging in these values into the formula, we get:
S8 = 12(1 - (1/2)^8) / (1 - 1/2)
Simplifying, we have:
S8 = (12 - 3/256) / (1/2)
Multiplying by the reciprocal, we get:
S8 = (12 - 3/256)(2/1)
S8 = (24 - 3/128)
Now we can convert 24 to a fraction with the same denominator:
S8 = (3072/128 - 3/128)
S8 = (3069/128)
Thus, the sum of the eighth term is 3069/128.