The sum of 25 consecutive integers is 1000. What is the smallest integer used as an addend.
the sum of the first n integers is
n(n+1)/2
Now, we have 25 integers starting somewhere else. That is, if the first one is k+1, then we add k to each value, giving us
nk + n(n+1)/2
So, using n=25,
25k + 25*26/2 = 1000
k + 13 = 40
k = 27
The numbers are 28,...,52
Checking our work, recall that the sum of n terms of an arithmetic sequence is
n(Ao+An)/2
= 25(28+52)/2 = 25*80/2 = 1000
Holy crap you know math
Let's assume the first integer used as an addend is "n".
The sum of 25 consecutive integers can be represented as:
n + (n+1) + (n+2) + ... + (n+23) + (n+24)
We are given that the sum of these integers is 1000. Therefore, we can write the equation:
n + (n+1) + (n+2) + ... + (n+23) + (n+24) = 1000
We can simplify this equation by combining like terms:
25n + (1+2+...+23+24) = 1000
25n + (24*25/2) = 1000
25n + 300 = 1000
Subtracting 300 from both sides of the equation:
25n = 700
Dividing both sides of the equation by 25:
n = 28
Therefore, the smallest integer used as an addend is 28.
To find the smallest integer used as an addend, we need to divide the total sum (1000) by the number of integers (25) to find the average. This average will be the middle term of the series.
Let's denote the smallest integer as 'x'. Since we have 25 consecutive integers, the next integer will be (x+1), and the next after that will be (x+2), and so on.
The sum of these consecutive integers can be calculated as follows:
Sum = (x) + (x+1) + (x+2) + ... + (x+23) + (x+24)
Since we have 25 terms, we can rewrite this as:
Sum = 25x + (1 + 2 + ... + 23 + 24)
The sum of the integers from 1 to 24 can be found using the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
Plugging in the values for this case:
sum = (24/2)(1 + 24)
= 12(25)
= 300
Now, we can rewrite the equation for the total sum:
1000 = 25x + 300
Simplifying the equation:
700 = 25x
Dividing both sides of the equation by 25:
x = 28
Therefore, the smallest integer used as an addend is 28.