Given the arithmetic sequence 11, 5, -1, ..., find .S(30)- on the bottom
You can easily see what a and d are, so
S30 = 30/2 (2a+29d)
Oh, you're asking for the sum of an arithmetic sequence, I see. Well, let's take a look at this interesting sequence of numbers! We have 11, 5, -1, and so on. Now, to find the sum of the first 30 terms, we need to find the common difference first.
By observing the sequence, we can see that each term is decreasing by 6. So, the common difference (d) is -6. Now that we have that, we can calculate the sum of the first 30 terms using the formula:
S(30) = (n/2)(2a + (n-1)d)
Here, n represents the number of terms, a is the first term, and d is the common difference.
For our sequence, considering the first term is 11, the common difference is -6, and we want to find the sum of the first 30 terms, we can substitute these values into the formula:
S(30) = (30/2)(2(11) + (30-1)(-6))
Now, let's get out our handy-dandy calculator and do some math... *calculating noises*
Ah! I've got it! The sum of the first 30 terms in this arithmetic sequence is -225. So, .S(30)- is -225 on the bottom. Quite a negative way to end, but hey, that's just how numbers roll sometimes!
To find the sum of the arithmetic sequence, we need to determine the common difference (d) and the first term (a₁).
In this sequence, the common difference (d) is calculated by subtracting the second term from the first term:
d = 5 - 11 = -6
The first term (a₁) is 11.
Now, we can use the formula to find the sum of the arithmetic sequence (Sₙ), where n is the number of terms:
Sₙ = (n/2)(2a₁ + (n-1)d)
Substituting the values into the formula:
S₃₀ = (30/2)(2(11) + (30-1)(-6))
Simplify and calculate:
S₃₀ = 15(22 + 29(-6))
S₃₀ = 15(22 - 174)
S₃₀ = 15(-152)
S₃₀ = -2280
Therefore, S₃₀ is equal to -2280.
To find the sum of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n-1)d
Where:
An is the nth term
A1 is the first term
n is the number of terms
d is the common difference
In the given sequence, the first term A1 is 11 and the common difference d is -6 (subtracting 6 from each term to get to the next term).
Using the formula for the nth term, we can find the 30th term:
A30 = 11 + (30-1)(-6)
A30 = 11 + 29(-6)
A30 = 11 - 174
A30 = -163
The 30th term of the sequence is -163.
Now, to find the sum of the sequence, we can use the formula for the sum of an arithmetic sequence:
Sn = (n/2)(A1 + An)
Where Sn is the sum of the first n terms.
Plugging in the values, we get:
S30 = (30/2)(11 + (-163))
S30 = 15(-152)
S30 = -2280
Therefore, the sum of the arithmetic sequence 11, 5, -1, ... up to the 30th term is -2280.