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The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

A. 10,000
B. 10,200
C. 15,050
D. 20,050
E. 20,100

PS85402.01

Originally posted by Rashed12 on 11 Mar 2019, 02:13.
Last edited by Bunuel on 21 Sep 2019, 02:47, edited 2 times in total.
Renamed the topic and edited the question.
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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Rashed12 wrote:
If the sum of the first 100 positive integers is 5050 then what is the sum of the first 200 positive integers?

A. 10,000
B. 10,200
C. 15,050
D. 20,050
E. 20,100

Various ways..
1) making use of the info in the question..
Sum of first 100 positive integers =5050. Now 101 to 200, each term will be 100 more than a certain term in 1 to 100. For example, 101 is 100 more than 1, 102 is again 100 more than 2... And so on till 200 is 100 more than 100.... Thus, Sum of 101 to 200 will be sum of 1 to 100+100*100=5050+10000=15050..
Sum of all 1 to 200 = 5050+15050=20,100.
2) Average
Average of 1 to 200 is middle of 100 and 101, so 100.50, thus sum = 200*100.50=20,100.
3) Sum = $$200*\frac{1+200}{2}$$=200*100.50=20,100.

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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Sum of first n consecutive numbers, = n/2(n+1)
SUm of first 10 consecutive numbers = 5*11 = 55
SUm of first 100 consecutive numbers = 50*101 = 5050 (Given)
SUm of first 200 consecutive numbers = 100*201 = 20,100 (Option E)
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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Rashed12 wrote:
The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

A. 10,000
B. 10,200
C. 15,050
D. 20,050
E. 20,100

PS85402.01

GIVEN: 1 + 2 + 3 + 4 + 5 + .....+ 99 + 100 = 5050

We want to find the sum 1 + 2 + 3 + 4 + 5 + .....+ 199 + 200
Notice that 1 + 2 + 3 + 4 + 5 + .....+ 199 + 200 = 1 + 2 + 3 + 4 + 5 + .....+ 99 + 100 + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + . . . (100 + 99) + (100 + 100)
= (1 + 2 + 3 + 4 + 5 + .....+ 99 + 100) + (1 + 2 + 3 + 4 + 5 + .....+ 99 + 100) + (100 + 100 + 100 + 100 + ...100 + 100) [ note that there are one hundred 100's in the sum]
= (5050) + (5050) + (100)(100)
= 10,100 + 10,000
= 20,100

Cheers,
Brent
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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Top Contributor
Rashed12 wrote:
The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

A. 10,000
B. 10,200
C. 15,050
D. 20,050
E. 20,100

PS85402.01

Another approach is to use the formula for the sum of the first positive n integers.
1 + 2 + 3 + 4 + .... + n = (n)(n+1)/2

So, the sum of the first 200 positive integers = (200)(200+1)/2
= (200)(201)/2
= (100)(201)
= 20,100

Cheers,
Brent
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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Hi All,

We're told that the sum of the first 100 positive integers is 5,050. We're asked for the sum of the first 200 positive integers. This question can be approached in a number of different ways - and since we're dealing with the sum of CONSECUTIVE integers, this question can be solved rather easily by BUNCHING.

To start, since we're dealing with the first 200 positive integers, we can form 100 'groups of 2'.... Let's start by adding the smallest and largest numbers....

1+200 = 201

Then the next smallest and next largest and so on...

2+199 = 201
3+198 = 201
Etc.

We'll end up with 100 'pairs' that equal 201, thus, the overall sum of all 200 numbers is (100)(201) = 20,100

GMAT assassins aren't born, they're made,
Rich
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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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Rashed12 wrote:
The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

A. 10,000
B. 10,200
C. 15,050
D. 20,050
E. 20,100

PS85402.01

We can use the formula: average x number = sum. To find the average, we add the first and last numbers and divide by 2. Thus, the sum of the first 200 integers is:

(1 + 200)/2 x 200 = 201 x 100 = 20,100

Alternate solution:

Since the sum of the first 100 positive integers (i.e., 1 to 100) is 5,050 and each of the next 100 positive integers (i.e., 101 to 200) is 100 more than its counterpart in the first 100 integers (101 is 100 more than 1, 102 is 100 more than 2, etc.), then the sum of the integers from 101 to 200 (inclusive) is 100 x 100 = 10,000 more than 5,050, i.e., 15,050. Therefore, the sum of the first 200 positive integers (i.e., 1 to 200) is 5,050 + 15,050 = 20,100.

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Re: The sum of the first 100 positive integers is 5,050. What is the sum  [#permalink]

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MBA HOUSE KEY CONCEPT: Summation of an arithmetic progression

Formula: (a1 + an) n / 2

a1 = first term = 1

an = last term = 200

n number of terms = 200

(1 + 200) 200 / 2 = 20100

E  Re: The sum of the first 100 positive integers is 5,050. What is the sum   [#permalink] 03 Jan 2020, 16:34
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