SOLUTION: If you add the consecutive counting numbers starting with 1, what number will cause the sum to exceed 1000? Explain.

Algebra.Com
Question 351050: If you add the consecutive counting numbers starting with 1, what number will cause the sum to exceed 1000? Explain.
Answer by edjones(8007)   (Show Source): You can put this solution on YOUR website!
n=number of terms. a[1]=1st term, a[n]=final term, S=sum of finite arithmetic sequence.
n/2(a[1]+a[n])=S[n]
n/2(1+n)>=1000
n(n+1)>=2000
n^2+n-2000>=0 Quadratic formula (below)
n>=44.22
n=45 the number that will cause the sum to exceed 1,000.
.
Ed
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=8001 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 44.2241545476267, -45.2241545476267. Here's your graph:
RELATED QUESTIONS

The product of three consecutive counting numbers is 24.If the biggest number is... (answered by Edwin McCravy)
if you add the digits in a number, how many numbers between 0 and 1000 will have a sum of (answered by Alan3354,greenestamps,ikleyn)
Given a starting number , you doubled it and add 1 , then divide the answer by 1 less... (answered by josgarithmetic)
What is the three consecutive numbers if there sum add up to 66 (answered by Alan3354)
if twice the sum of three consecutive numbers is 12, and the two lowest numbers add up to (answered by Alan3354)
sum of Three odd consecutive number exceed the smallest by 36.Find the numbers? (answered by CubeyThePenguin)
If the sum of two counting numbers is an even counting number, then the product of the... (answered by Alan3354,josgarithmetic)
could you help me? thank, numbers that are in order and differ by 1, such as 2,3,4 are... (answered by edjones)
If you add a prime number to itself,is the sum composite or prime?... (answered by solver91311)