Answer:
Therefore, the tenth term in the series is (-1/6) * (-3)^(n-1).
Explanation:
AI-generated answer
Apologies for the confusion in the previous response. The series you provided is not a geometric series since the terms do not have a common ratio.
To find the sum of the first 10 terms in the series (-1/6) + (1/-4) + (-3/8) + (-9/16) + ..., we can calculate each term and then add them up.
The given series follows a specific pattern, where each term is obtained by multiplying the previous term by a negative power of 3.
Let's calculate the first 10 terms:
1st term = -1/6
2nd term = (1/-4) * (-1/3) = 1/12
3rd term = (-3/8) * (-1/3)^2 = -1/24
4th term = (-9/16) * (-1/3)^3 = 1/48
...
Continuing this pattern, we can calculate the remaining terms until the 10th term.
Now, let's add up the first 10 terms:
-1/6 + 1/12 + (-1/24) + 1/48 + ...
To simplify the calculations, we can find a common denominator for all the fractions. In this case, the common denominator is 48.
-1/6 can be written as -8/48, 1/12 as 4/48, -1/24 as -2/48, and so on.
-8/48 + 4/48 - 2/48 + ...
Now, we can simply add up the numerators:
(-8 + 4 - 2 + ...) / 48
The numerators form a pattern of -8, 4, -2, 1, -1, 1, -1, ...
The sum of this pattern is -5. Therefore, the numerator of the sum is -5, and the denominator remains 48.
So, the sum of the first 10 terms in the series is -5/48.
To find the nth term, we can observe that each term follows the pattern of multiplying the previous term by a negative power of 3. Therefore, the nth term can be calculated using the formula:
an = a * (-3)^(n-1),
where an represents the nth term, a is the first term, and n is the term number.
In this case, the first term a is -1/6.
Using the formula, we can find the nth term:
an = (-1/6) * (-3)^(n-1)
Therefore, the nth term in the series is (-1/6) * (-3)^(n-1).