Answer: 528
Explanation:
The expression ∑n=1^24 (2n−3) represents the sum of the expression (2n−3) as n takes on values from 1 to 24. To calculate this sum, you can use the formula for the sum of an arithmetic series:
∑n=1^24 (2n−3) = [n(n+1)/2] * (2n−3)
Now, plug in the values for n from 1 to 24 and calculate each term in the sum:
= [1(1+1)/2] * (2(1)−3) + [2(2+1)/2] * (2(2)−3) + ... + [24(24+1)/2] * (2(24)−3)
Now, compute each term:
= [1(2)/2] * (2−3) + [2(3)/2] * (4−3) + ... + [24(25)/2] * (48−3)
= (1) * (-1) + (3) * (1) + (6) * (3) + ... + (300) * (45)
Now, sum up all these individual terms:
= -1 + 3 + 18 + ... + 13500
To find this sum, you can use the formula for the sum of an arithmetic series:
Sum = (n/2) * [2a + (n-1)d]
In this case, a is the first term (-1), n is the number of terms (24), and d is the common difference (2).
Sum = (24/2) * [2(-1) + (24-1)(2)]
Now, calculate this sum:
Sum = 12 * [-2 + 46]
Sum = 12 * 44
Sum = 528
So, ∑n=1^24 (2n−3) is equal to 528.