Consider the sequence
1,-2, 3,-4, 5, -6,.........., n(-1)^(n+1)
What is the average of first 300 terms of the sequence?
rearrange the terms to see that the first 2n terms are
1+3+5+... - 2(1+2+3+...)
= n^2 - 2(n(n+1)/2)
= n^2 - n^2-n
= -n
so, the average is -n/2n = -1/2
or, rearrange to see that the first 2n terms are
(1-2)+(3-4)+...+(n-(n+1))
= -1 + -1 + ...
= -n
average is -n/2n = -1/2
To find the average of the first 300 terms of the sequence, we first need to understand the pattern of the sequence.
Looking at the given sequence, we can observe that it alternates between positive and negative terms. The positive terms start with 1, 3, 5, and so on, while the negative terms start with -2, -4, -6, and so on.
We can also notice that the sign of each term alternates based on the position in the sequence. The odd-numbered terms (1st, 3rd, 5th, etc.) are positive, while the even-numbered terms (2nd, 4th, 6th, etc.) are negative.
Knowing this pattern, we can determine that the n-th term of the sequence is given by: n(-1)^(n+1).
To find the average of the first 300 terms, we need to add up all the terms and then divide the sum by the total number of terms, which in this case is 300.
Let's calculate the average step by step:
1. Calculate the sum of the first 300 terms:
Since we know the pattern of the alternation, we can split the sum into two parts: one for the positive terms and one for the negative terms.
For the positive terms (odd-numbered terms):
To find the sum of the positive terms, we add up all the odd numbers from 1 to 599 (300 odd numbers since it's the first 300 terms of the sequence). We can use the formula for the sum of an arithmetic series:
sum_of_positive_terms = (n/2) * (first_term + last_term)
where n is the number of terms and first_term is the first term of the sequence.
sum_of_positive_terms = (300/2) * (1 + 599) = 150 * 600 = 90,000
For the negative terms (even-numbered terms):
To find the sum of the negative terms, we add up all the even numbers from -2 to -600 (300 even numbers since it's the first 300 terms of the sequence). Again, we can use the formula for the sum of an arithmetic series:
sum_of_negative_terms = (n/2) * (first_term + last_term)
To calculate the first_term and last_term, we can plug in the values of n and find the corresponding terms of the sequence:
first_term = 2 * (1) = 2
last_term = 2 * 300 = 600
sum_of_negative_terms = (300/2) * (2 - 600) = 150 * (-598) = -89,700
2. Add up the sum of the positive terms and the sum of the negative terms:
total_sum = sum_of_positive_terms + sum_of_negative_terms = 90,000 + (-89,700) = 300
3. Calculate the average:
average = total_sum / number_of_terms = 300 / 300 = 1
Therefore, the average of the first 300 terms of the sequence is 1.