a. what is the general term of the following series?
60/121-30/11+15-... +219 615/16
b. how many terms does the series have?
- 15 x ² + 2x + 13 = 0
46
math art wrighting spelling
a. To find the general term of the series, we need to examine the pattern and look for a common rule. Let's break down the given series:
60/121 - 30/11 + 15 - ... + 219615 / 16
By analyzing the terms, we can notice that each term follows a specific pattern. We can identify three sequences in the series: one for the numerators, one for the denominators, and one for the signs:
Numerators: 60, 30, 15, ...
Denominators: 121, 11, 1, ...
Signs: -, +, -, ...
Looking at the numerators, we can see that each term is halved from the previous term. This can be represented as a geometric sequence with a common ratio of 1/2.
Looking at the denominators, we see that each term is squared and then reduced by 1. This can be represented as an arithmetic sequence with a common difference of -110.
The signs alternate between positive and negative with each term. We can use the pattern (-1)^(n-1) to represent the signs, where n is the position of the term.
Combining these observations, we can write the general term of the series as:
((-1)^(n-1)) * (60 * (1/2)^(n-1)) / (11^(2n-1))
b. To determine the number of terms in the series, we need to find out the position of the last term. Let's first examine the pattern of the series.
By analyzing the series, we can see that each term decreases in magnitude as the series progresses. Therefore, the last term is the most significant term, and we need to figure out its position in the series.
The series starts with 60/121 and ends with 219615/16, with each term being progressively smaller in magnitude. To find the position of the last term, we need to determine how the terms are changing.
Considering the numerators, we can see that each term is halved from the previous term. This indicates a possibility of a geometric sequence with a common ratio of 1/2.
To find the number of terms, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1),
where a1 denotes the first term, r denotes the common ratio, and n denotes the position of the term.
In the given series, the first numerator is 60. So using the formula:
219615/16 = 60 * (1/2)^(n-1)
We can solve this equation to determine the value of n.
To simplify, we can rewrite 219615/16 and 60 with the same base:
(1/2)^(n-1) = (219615/16) / (60)
By taking the logarithm of both sides, we can solve for n:
(n-1) * log(1/2) = log((219615/16) / (60))
Solving for n will give us the position of the last term, and hence the number of terms in the series.