what is the general term of the following series?
60/121-30/11+15-...+219 615/ 16
To find the general term of the given series, we need to first identify the pattern. Looking at the series, we can observe that each term alternates between addition and subtraction.
Let's break down the series:
60/121 - 30/11 + 15 - ... + 219615/16
If we examine the denominators, we see that they follow a pattern where each denominator is getting multiplied by 11:
121, 11, 1, ...
We can obtain the next denominator by multiplying the previous denominator by 11:
1 * 11 = 11
11 * 11 = 121
Following this pattern, we can find the general term for the denominator in the series as:
Denominator = 121 / 11^(n-1)
Where n represents the position of the term in the series.
Next, let's analyze the numerator of each term:
60, -30, 15, ..., 219615
Observing the numerators, we can notice that they are obtained by multiplying the previous numerator by -1 and then multiplying by 2 in each alternate term:
60 * -1 * 2 = -120
-120 * -1 * 2 = 240
And so on...
We can derive the general term for the numerator in the series as:
Numerator = (-1)^(n-1) * 30 * 2^(n-1)
Now, combining the numerator and denominator, we can obtain the general term of the series:
General Term = (-1)^(n-1) * 30 * 2^(n-1) / (121 / 11^(n-1))
Where n represents the position of the term in the series.