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Veritas Prep GMAT Instructor Joined: 11 Dec 2012
Posts: 312
Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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Hi neerajeai, please note that the n(n+1)/2 shortcut formula is only applicable if the starting point is 1. Anytime you want to find the number of terms between two given numbers you should use the general formula ((first - last) / frequency) + 1. You can also multiply by the average at the end to get the sum.

In your case it is (((301-99)/2) + 1) * 200 = 102 * 200 = 20400. Answer choice C with a presumed typo in the unit digit?

Hope this helps!
-Ron
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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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gwiz87 wrote:
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 even integers between 99 and 301?

A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150

Hi,

Hoping you can explain this one to me.

Although a formula is provided in this problem, we can easily solve it using a different formula:

sum = (average)(quantity)

Let’s first determine the average.

In any set of numbers in an arithmetic sequence, we can determine the average using the formula:

(1st number in set + last number in set)/2

Remember, we must average the first even integer in the set and the last even integer in the set. So we have:

(100 + 300)/2 = 400/2 = 200

Next we have to determine the quantity. Once again, we include the first even integer in the set and the last even integer in the set. Thus, we are actually determining the quantity of even consecutive even integers from 100 to 300, inclusive.

Two key points to recognize:

1) Because we are determining the number of “even integers” in the set, we must divide by 2 after subtracting our quantities.

2) Because we are counting the consecutive even integers from 100 to 300, inclusive, we must “add 1” after doing the subtraction.

quantity = (300 – 100)/2 + 1

quantity = 200/2 + 1

quantity = 101

Finally, we can determine the sum.

sum = 200 x 101

sum = 20,200

Note that the reason this is easier than using the formula provided, is that the given formula would have to be applied several times since we don’t want the total of all of the first 300 numbers. We’d have to remember to subtract the sum of the first 99 and divide by two to count only the even numbers. But we’d also have to account for the fact that the first and the last number in the set are both even. So even though a formula is given, it isn’t very easy to use.

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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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niks18

I was confused if the second formula was applied only when my sequence is starting from 1.
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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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niks18

I was confused if the second formula was applied only when my sequence is starting from 1.

The formula is a simple derivation of AP series that starts from 1 and is consecutive

So the Sum of AP series is $$S_n=\frac{n[2a+(n-1)d]}{2}=\frac{n}{2}$$[First term + Last Term]

if the sequence starts from 1 and is consecutive so your First term=1 and Last Term=number of terms in the series (because sequence is consecutive)

so $$S_n=\frac{n(1+n)}{2}$$

Let's take a simple example. if I have to find sum of numbers from 6 to 10, then I can do that by first finding sum of all Numbers from 1 to 10 and then from this sum subtract sum of numbers from 1 to 5.

In the number line this can be represented as

1...2...3...4...5...6...7...8...9...10. Here First term =1 and last term=10=number of terms in the sequence

As you can clearly see to find sum of numbers from 6 to 10, we simply need to remove sum of numbers from 1 to 5 out of the total sum i.e from 1 to 10.

Now in this question we need to find sum of number from 50 to 150 so it will be Total sum of numbers from 1 to 150 less sum of numbers from 1 to 49
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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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gwiz87 wrote:
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 even integers between 99 and 301?

A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150

Here's one approach.

We want 100+102+104+....298+300
This equals 2(50+51+52+...+149+150)
From here, a quick way is to evaluate this is to first recognize that there are 101 integers from 50 to 150 inclusive (150 - 50 + 1 = 101)

To evaluate 2(50+51+52+...+149+150), let's add values in pairs:

....50 + 51 + 52 +...+ 149 + 150
+150+ 149+ 148+...+ 51 + 50
...200+ 200+ 200+...+ 200 + 200

How many 200's do we have in the new sum? There are 101 altogether.
101 x 200 = 20,200

Approach #2:

From my last post, we can see that we have 101 even integers from 100 to 300 inclusive.

Since the values in the set are equally spaced, the average (mean) of the 101 numbers = (first number + last number)/2 = (100 + 300)/2 = 400/2 = 200

So, we have 101 integers, whose average value is 200.
So, the sum of all 101 integers = (101)(200)
= 20,200
= B

Approach #3:

Take 100+102+104+ ...+298+300 and factor out the 2 to get 2(50+51+52+...+149+150)
From here, we'll evaluate the sum 50+51+52+...+149+150, and then double it.

Important: notice that 50+51+.....149+150 = (sum of 1 to 150) - (sum of 1 to 49)

Now we use the given formula:
sum of 1 to 150 = 150(151)/2 = 11,325
sum of 1 to 49 = 49(50)/2 = 1,225

So, sum of 50 to 150 = 11,325 - 1,225 = 10,100

So, 2(50+51+52+...+149+150) = 2(10,100) = 20,200

Cheers,
Brent
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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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We are given the formula:

$$∑_1^n \frac{n(n+1)}{2}$$

We're asked to find sum of all the even numbers between $$99$$ and $$301$$....

$$100+102+104+...+296+298+300$$

We can factor out $$2$$ from this sum:

$$2 \times (50+51+52+...+148+149+150)$$

In the brackets we have sum of numbers from $$50$$ to $$150$$. We can use the given formula for the sum of $$1$$ to $$150$$ and subtract the sum of $$1$$ to $$49$$ to find the total of this sum.

$$2 \times (\frac{150 \times 151}{2}-\frac{49 \times 50}{2})$$

$$=(150)(151) – (49)(50)$$

$$=22,650 – 2,450$$

$$=20,200$$

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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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CharmWithSubstance wrote:
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 even integers between 99 and 301?

A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150

So I know, that we must use the first and last digits of even numbers, if we are calculating for a sum of evens..

But I tried it this way, and still got the answer..
Would like to know, if it's an acceptable method, or its wrong, and I shouldn't use it ?

if we use first and last terms as given, i.e. First term= 99 and last term = 301
Subtract these two numbers, to get the total number of terms..
301-99 = 202
Note this includes odd and evens both

for even, we must divide it by 2.
therefore, we get a 202/2 = 101 terms...

AS you can see, nothing is added or subtracted for inclusive, exclusive..

now, 101 * avg of terms, will give us the sum.

avg = 301+99/2 = 400/2 = 200

101*200 = 20,200
option B

Please correct me, if you think this is a wrong approach and we should not use it
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For any positive integer n, the sum of the first n positive integers  [#permalink]

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This is another way to tackle this question:

It is given that the sum of the first n positive integers is n(n+1)/2. I didn't really use this formula.

I remembered this formula instead (and I think you should remember it too!):

The sum of the elements in any evenly spaced set is: (mean)*(# of terms)

Steps:
1. First, find the mean
The questions asked to find the sum of all the EVEN integers between 99 and 301. So that means we need to look at EVEN numbers only. So really the range we are looking at is between 100 and 300. So the mean of these evenly spaced numbers between 100 and 300 is 200. --> this is the mean (no calculation necessary here because it's an evenly spaced range so its very straight forward to find the mean; just get the middle number!)

2. Second, find the # of EVEN terms.
We stated in step 1 that the EVEN number range is between 100 and 300. So here we can utilise this formula:
[(last term -first term)/2] +1

which comes to:
[(300-100)]/2 + 1
= (200/2) +1
= 100+1
= 101 --> this is the # of EVEN terms in the range 100 and 300

Coming back to this: The sum of the elements in any evenly spaced set is: (mean)*(# of terms)

(mean)*(# of terms)
=200* 101
=20200

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Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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In my case I remember the formula of arithmetic progression: Sum of n terms = n/2*(a1+an)
In this case:

Interval: 100-102-104-106........-296-298-300

The difference (distance) between each number is 2:
- If I do (102-100)/2=1
- If I do (104-100)/2=2
- If I do (106-100)/2=3
As we can see, if we do this, every term is in the (n-1) position... so (300-100)/2=100 will be in the n-1 position, then... the number 300 is in the position (100+1=101) of the serie.

Now we apply the formula: Sum of n = n/2*(a1+an)
Sum of n = 101/2*(100+300)=20.200

I Hope this solution could help someone

Thanks!
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Posts: 27
Re: For any positive integer n, the sum of the first n positive integers  [#permalink]

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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 even integers between 99 and 301?

A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150

To find the solution, treat this like any other consecutive integer problem. To find the sum of any consecutive integers, multiply the average of those integers times the number of integers.

So first, to find the average of the integers, we know that the lower limit is 100 and the upper limit is 300, since we are only looking for even integers. To find the average, (300 + 100) / 2, which gives us 200.

Next, find the total number of integers. We know that 100 is the 50th even integer and that 300 is the 150th even integer. We subtract 150 - 50 to get 100 even integers, but must add 1 to the total since we are looking for an INCLUSIVE number.

So, we know the average of the integers in this range is 200 and the number of integers in the range is 101. Follow the principles of consecutive integers to find the answer: SUM = (avg) ( # of integers).

This gets us 20,200 = 200 * 101. Re: For any positive integer n, the sum of the first n positive integers   [#permalink] 19 Mar 2020, 13:06

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