Find the sum of the first 33 terms of the arithmetic sequence whose first term is 3 and whose 33rd term is -253.
If you are working on arithmetic sequence, I assume you know this equation.
x = a + d(n-1)
Okay, so the first thing to do is to find the common difference.
-253 = 3 + d(33-1)
Solve for "d" and you get
d = -8
Next, I also assume you know the equation for the sum of an arithmetic sequence.
(n/2)(2a + (n-1)d)
So, plug the numbers in the equation and...
sum = (33/2)(2 x 3 - 8(33-1))
sum = -4125
given:
a = 3
term(33) = a+32d = -253
32d = -256
d = -8
sum(33) = (33/2)(first + last)
= (33/2)(3 -253) = -4125
To find the sum of an arithmetic series, we need to use the formula:
Sn = (n/2)(a + l),
where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term (a) is 3 and the 33rd term (l) is -253.
We know that the sum of the first 33 terms (Sn) is needed, so n = 33.
Substituting these values into the formula, we get:
Sn = (33/2)(3 + (-253))
Simplifying inside the parentheses:
Sn = (33/2)(-250)
Now we need to calculate the value of Sn:
Sn = (33/2)(-250) = -4125
To find the sum of the first 33 terms of an arithmetic sequence, you can use the formula for the sum of an arithmetic sequence.
The formula is given as:
S = (n/2)(2a + (n-1)d)
where:
S = the sum of the terms
n = number of terms
a = first term
d = common difference
In this case, we are given that the first term (a) is 3, the 33rd term is -253, and we need to find the sum of the first 33 terms. We can plug in these values into the formula and solve for S.
So, let's calculate step by step:
n = 33 (since we need the sum of the first 33 terms)
a = 3 (the first term of the sequence)
d = ? (we need to find the common difference)
To find the common difference (d), we can use the formula for the nth term of an arithmetic sequence:
nth term = a + (n-1)d
Plugging in the values, we have:
-253 = 3 + (33-1)d
Simplifying, we get:
-253 = 3 + 32d
-253 - 3 = 32d
-256 = 32d
d = -256/32
d = -8
Now that we have the value of d, we can substitute it back into the original formula
S = (n/2)(2a + (n-1)d)
S = (33/2)(2(3) + (33-1)(-8))
Simplifying further:
S = (33/2)(6 - 264)
S = (33/2)(-258)
S = -8,514
Therefore, the sum of the first 33 terms of the given arithmetic sequence is -8,514.