Find the sum of the 50 greatest negative integers.
I'm supposed to find a, d, and t(at)n, but I'm not sure which is which. I think it makes sense for n to equal 50, but apart from that, I don't know what else to do. I would really appreciate some sort of explanation on how to solve this. Thank you. Oh, and I have a test on this tomorrow, so I really need the help. :)
you are simply asked to find
(-50) + (-49) + ... + (-2) + (-1)
there are 50 of these, so n=50
the first term is -50, so a=-50
the common difference is 1, so d=1
S50 = 50/2(-50 + (-1) = -1275
I was using S(n) = n/2(first + last)
You could have used
S(n) = n/2[2a + (n-1)d]
= 25[-100 + 49]
= -1275
notice if you had added up the first 50 positive integers you would get +1275
oh! a = -50?!
for some reason I thought it was 1, but it makes much more sense now. Okay, yeah, I see what you mean. :) Thank you so very much for helping me! :D
To solve this problem, we need to identify the given information and the formula for finding the sum of an arithmetic series.
In an arithmetic series, we have the following formula:
Sn = (n/2) * (2a + (n-1)d)
Where:
- Sn is the sum of the series
- n is the number of terms in the series
- a is the first term of the series
- d is the common difference between the terms
Given that we want to find the sum of the 50 greatest negative integers, we can conclude that "n" refers to the number of terms, which is 50.
Since we are dealing with negative integers, the common difference (d) will be -1 since each term is decreasing by 1.
To determine the first term (a), we should find the largest negative integer in the series of 50 integers. The largest negative integer will be found at the beginning of the series. So, the first term (a) will be the largest negative integer.
The formula for the sum of an arithmetic series becomes:
Sn = (n/2) * (2a + (n-1)d)
Substituting the given values:
- n = 50 (number of terms)
- a = -1 (first term)
- d = -1 (common difference)
Sn = (50/2) * (2 * (-1) + (50-1) * (-1))
Simplifying the equation:
Sn = 25 * (-2 + 49 * (-1))
Sn = 25 * (-2 - 49)
Sn = 25 * (-51)
Sn = -1275
The sum of the 50 greatest negative integers is -1275.
Remember to double-check the values used and all calculations to ensure accuracy before submitting your final answer. Good luck with your test! Let me know if there's anything else I can help you with.
To find the sum of the 50 greatest negative integers, let's break it down step by step.
First, let's determine what each variable represents in this problem.
- "a" represents the first term in the sequence.
- "d" represents the common difference between consecutive terms in the sequence.
- "t(at)n" represents the nth term in the sequence.
In this case, we're dealing with negative integers, so the first term "a" would be the largest negative integer, for example, -1. The common difference "d" between consecutive terms would be -1 since the numbers are descending by 1, and "t(at)n" refers to the 50th term in the sequence.
With this information, we can now proceed to find the sum.
Step 1: Determine the first term (a) and the common difference (d).
Since we are dealing with negative integers, the first term (a) would be the largest negative integer. So, in this case, the first term (a) would be -1. The common difference (d) between consecutive terms in the sequence is -1.
Step 2: Find the nth term (t(at)n).
We know that the nth term (t(at)n) is the 50th term in the sequence. Since the common difference is -1, we can use the formula for the nth term of an arithmetic sequence:
t(at)n = a + (n - 1) * d
Substituting the values, we get:
t(at)50 = -1 + (50 - 1) * -1
= -1 + 49 * -1
= -1 - 49
= -50
Therefore, the 50th term (t(at)50) in this sequence is -50.
Step 3: Find the sum of the sequence using the formula for the sum of an arithmetic sequence.
The formula to find the sum of an arithmetic sequence is:
Sn = (n / 2) * (2a + (n - 1) * d)
Substituting the values, we get:
S50 = (50 / 2) * (2 * -1 + (50 - 1) * -1)
= (25) * (-2 + 49 * -1)
= 25 * (-2 - 49)
= 25 * -51
= -1275
Therefore, the sum of the 50 greatest negative integers is -1275.
Remember to double-check the values of "a" and "d" based on the given problem statement. Good luck on your test!