How many terms of the series -6+0+6+12+... are needed to yield a sum of 324?
To find out how many terms are needed to yield a sum of 324, we need to determine the pattern in the series and calculate the sum.
First, let's observe the pattern in the series: -6, 0, 6, 12, ...
We can see that each term is increasing by 6. This means the series follows an arithmetic sequence with a common difference (d) of 6.
Next, let's find the formula for the nth term of the arithmetic sequence. The formula for the nth term (Tn) of an arithmetic sequence is given by:
Tn = a + (n-1)d
In our case, the first term (a) is -6, and the common difference (d) is 6. Plugging these values into the formula:
Tn = -6 + (n-1)6
Tn = -6 + 6n - 6
Tn = 6n - 12
Now, we can find the sum of the series up to the nth term using the formula for the sum of an arithmetic sequence:
Sn = (n/2)(2a + (n-1)d)
In our case, we want to find the sum (Sn) equal to 324. Plugging in the values:
324 = (n/2)(2 * -6 + (n-1) * 6)
324 = (n/2)(-12 + 6n - 6)
324 = (n/2)(6n - 18)
Let's simplify this equation:
324 = (3n/2)(2n - 6)
324 = 3n^2 - 18n
Rearranging and dividing by 3:
3n^2 - 18n - 324 = 0
n^2 - 6n - 108 = 0
This quadratic equation can be factored as:
(n - 12)(n + 9) = 0
Setting each factor equal to 0:
n - 12 = 0 or n + 9 = 0
Solving for n:
n = 12 or n = -9
Since the number of terms (n) cannot be negative, we discard n = -9 as an extraneous solution.
Therefore, we conclude that the number of terms needed to yield a sum of 324 is 12.