For any positive integer n, the sum of the first n positive integers

Question

For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the integers between 250 and 350, inclusive?

(A) 100
(B) 6,600
(C) 20,200
(D) 30,300
(E) 55,000

Chasingthesun · Answer

to find sum , using formula : sum = Avg*qty 
average= (350+250)/2 = 300
qty = (350-250) + 1 ( inclusive) = 101 
therefore, sum = 300*101

Option D

sych · Answer

any alternative answer to this question?

Krunaal · Answer

An alternative method is to find sum of 350 digits and subtract sum of 249 digits =>  \frac{350*351}{2} - \frac{249*250}{2}  ; but this will leave you with a more complex multiplication than the other method viz. Avg * no. of terms =>  \frac{350*250}{2} * 101

sych · Answer

Okayy. So what would be the best solution ?

GraemeGmatPanda · Answer

That depends on what you're more comfortable with! Most people prefer the first one. Visualising what goes on helps to understand the maths:

First write out the first few numbers and the last few numbers in the sum:

Sum = 250 + 251 + 252 + ... + 348 + 349 + 350

If you were to calculate the average of all those numbers you would do:
Sum / number of terms = average

so if you knew the average you would be able to work out the sum by doing:
Sum = average x number of terms

1/ Now to work out the average, you can pair up the numbers:
If you pair up the first and the last: 250 + 350 = 600 -- > their average is 600/2 = 300
If you pair up the second and the second last: 251 + 349 = 600 -- > their average is 600/2 = 300
If you pair up the third and the third last: 252 + 348 = 600 -- > their average is 600/2 = 300
etc

There is one number in the middle that doesnt have a pair, but it's 300, so again the same average

For this reason the whole average is 300

2/ Then you need calculate the number of terms by doing last - first + 1

number of terms = 350 - 250 + 1 = 101

The +1 comes because you want the numbers 'inclusive'

So now you can apply the formula Sum = average x number of terms = 300 x 101 = 30,300

Answer D

Hope that helps!

Final note: If you're good at divisibility, once you've worked out the average (300), you know that the answer will be divisible by 300. This means it will be divisible by 3, which rules out A, C and E. And you could probably guess that B is too low.

mohitchowdhury · Answer

Use AP formula to find sum with common difference as d=1
S=n/2(2a+(n-1)d) Here n=101 and d=1 and a=250

For any positive integer n, the sum of the first n positive integers

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For any positive integer n, the sum of the first n positive integers

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