Set M consists of the consecutive integers from -15 to y, inclusive. If the sum of all the integers in set M is 70, How many numbers are in the set?
a. 33
b. 34
c. 35
d. 36
e. 37
To find out the number of integers in set M, we need to determine the range of integers from -15 to y that sum up to 70.
The sum of integers in set M can be calculated using the formula for the sum of an arithmetic series:
Sum = (N/2) * (first term + last term),
where N is the number of terms, and the first term and the last term denote the first and last numbers in the series, respectively.
In this case, the first term is -15, and the last term is y.
We are given that the sum of all the integers in set M is 70. So we can set up the equation:
70 = (N/2) * (-15 + y).
Now let's solve for y. Multiply both sides by 2:
140 = N * (-15 + y).
Next, distribute:
140 = -15N + Ny.
Rearrange the equation:
Ny - 15N = 140.
Factor out N:
N(y - 15) = 140.
Now, we need to factorize 140 to find its divisors. The factors of 140 are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
We can substitute these factors into the equation for N(y - 15) = 140 to find values for N and y. Let's check each factor:
For N = 1, 1(y - 15) = 140. The only solution for y in this case is y = 155.
For N = 2, 2(y - 15) = 140. The only solution for y in this case is y = 85.
For N = 4, 4(y - 15) = 140. The only solution for y in this case is y = 55.
For N = 5, 5(y - 15) = 140. The only solution for y in this case is y = 47.
For N = 7, 7(y - 15) = 140. The only solution for y in this case is y = 35.
For N = 10, 10(y - 15) = 140. The only solution for y in this case is y = 29.
For N = 14, 14(y - 15) = 140. The only solution for y in this case is y = 25.
For N = 20, 20(y - 15) = 140. The only solution for y in this case is y = 20.
For N = 28, 28(y - 15) = 140. The only solution for y in this case is y = 17.
For N = 35, 35(y - 15) = 140. The only solution for y in this case is y = 16.
For N = 70, 70(y - 15) = 140. The only solution for y in this case is y = 17.
From these solutions, we can see that the pair (N, y) is (70, 17).
Therefore, there are 70 numbers in the set M.
To find the number of integers in set M, we need to determine the value of y.
Let's calculate the number of integers between -15 and y, inclusive:
There are 15 integers from -15 to -1 (including both endpoints).
There is an additional integer for y.
So, the number of integers in set M is 15 + 1 = 16.
Next, let's set up the equation to solve for y:
-15 + (-14) + (-13) + ... + y = 70
To simplify the expression, we can group the consecutive integers from -15 to y:
(-15 + (-14) + (-13) + ...) + y = 70
To find the sum of the consecutive integers, we can use the formula for the sum of an arithmetic series:
Sum = (n / 2) * (first term + last term)
In this case, the first term is -15, and the last term is y.
Substituting these values into the equation, we get:
[(n / 2) * (-15 + y)] + y = 70
Now, let's solve for y:
[(16 / 2) * (-15 + y)] + y = 70
8 * (-15 + y) + y = 70
-120 + 8y + y = 70
9y - 120 = 70
9y = 190
y = 190 / 9
y ≈ 21.11
Since y represents an integer in this problem, we can round it up or down to the nearest whole number. Therefore, y = 21.
Thus, the number of integers in set M is 16.
-15 to +15 includes 31 integers, counting zero. The sum is zero.
so 16+17+18+19=70 (ie, 17.5 x 4).
so I count 31+5 integers.