The sum of the first 18 terms of an arithmetic series is 576. The common difference is -8. Determine the value of the first term.
Let the first term of the arithmetic series be $a$. We know that the sum of the first $18$ terms is $576$, so the sum of the first $17$ terms is $576-18a$. Now let's find the sum of the first $17$ terms in two ways.
First, we can find the sum of the first $17$ terms directly. The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so this sum is $$\dfrac{17(a+(a+(-8\cdot 17)))}{2} = 576-18a.$$Second, we can also find the sum of the first $17$ terms by finding the sum of the first $18$ terms, and then subtracting out the $18^\text{th}$ term $a+(-8\cdot 17) = a-136$. The sum of the first $17$ terms is then $$576 - (a-136) = 712 - a.$$Since we found the sum of the first $17$ terms in two different ways, we must have $$\begin{aligned}576 - 18a&= 712 - a\\\Rightarrow\qquad 17a&=136\\\Rightarrow\qquad a&=\boxed{8}.\end{aligned}$$
To find the value of the first term in an arithmetic series, we can use the formula for the sum of an arithmetic series:
Sn = n/2 * (2a + (n-1)d)
where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Given that Sn = 576, d = -8, and n = 18, we can substitute these values into the formula:
576 = 18/2 * (2a + (18-1)(-8))
Simplifying the right side of the equation:
576 = 9 * (2a - 17 * 8)
576 = 9 * (2a - 136)
Divide both sides of the equation by 9:
64 = 2a - 136
Rearranging the equation:
2a = 64 + 136
2a = 200
Divide both sides of the equation by 2:
a = 100
Therefore, the value of the first term in the arithmetic series is 100.
To find the value of the first term, you can use the formula for the sum of an arithmetic series, which is:
Sn = n/2 * (2a + (n-1)d)
where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
In this case, you have the following information:
- Sn = 576
- d = -8
- n = 18
Plugging in these values into the formula, we get:
576 = 18/2 * (2a + (18-1)(-8))
Simplifying the equation:
576 = 9 * (2a + 17(-8))
576 = 9 * (2a - 136)
576 = 18a - 1224
18a = 1800
a = 100
Therefore, the value of the first term is 100.