What is the sum of the first 100 positive odd numbers?

Question

A. 5,000 
B. 7,500 
C. 8,000 
D. 10,000 
E. 12,000

keats · Accepted Answer

For first n odd numbers, its n^2 and for first n even numbers, its n (n+1)

Here, it'll be (100)^2 = 10000

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Buzz93 · Answer

This one is about pattern recognition. Set up a table as follows:
 
Number Sum
 1 1
 3 4
 5 9
 7 16
 9 25

Now as you see, the sum always equals the perfect square of the number of odd integers that you want to sum. So the sum of the first 5 odd integers is 5^2 = 25. The pattern continues. Thus, the sum of the first 100 odd integers will be equal to 100^2 = 10,000.  The correct answer is (D).

BrentGMATPrepNow · Answer

APPROACH #1 : Look for a pattern
Sum of the first 2 odd numbers = 1 + 3 =  4 
Sum of the first 3 odd numbers = 1 + 3 + 5 =  9 
Sum of the first 4 odd numbers = 1 + 3 + 5 + 7 =  16 
Sum of the first 5 odd numbers = 1 + 3 + 5 + 7 + 9 =  25 
Aha!

Notice that: 
Sum of the first  2  odd numbers =   2 ² 
Sum of the first  3  odd numbers =   3 ² 
Sum of the first  4  odd numbers =   4 ² 
Sum of the first  5  odd numbers =   5 ² 
In general, the sum of the first  n  odd numbers =   n ²

So, the sum of the first  100  odd numbers =   100 ²  =  10,000

Answer: D

APPROACH #2 : Apply how useful formula
Before answering any GMAT quant problem,  always check the answer choices first 
Here, the answer choices are somewhat spread apart, which means we can be  somewhat aggressive  in our estimations.

Nice formula: 1 + 2 + 3 . . . + n = (n)(n + 1)/2

Let's use the above formula to find the sum of the first 200 integers (including odds AND evens)
1 + 2 + 3 . . . + 199 + 200 = (200)(200 + 1)/2
= (200)(201)/2
= (100)(201)
= 20,100
So, the sum of the first 200 integers is 20,100
HALF of those integers are ODD and HALF are even. So, this sum includes the sum of the first 100 ODD integers and the sum of the first 100 EVEN integers. 
So, the sum of the first 100 ODD integers is APPROXIMATELY 20,100/2 
20,100/2 = 10,050
So, the sum of the first 100 ODD integers ≈ 10,050
Answer choice D is the only one that's close to 10,050 so it must be the correct answer.

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MathRevolution · Answer

==> Since 1+3+5+….+(2n-1)=n 2, 1+3+…….(2*100-1)=100 2 =10,000 is derived.

Hence, the answer is D.

koenh · Answer

The sum of an arithmetic sequence equals (n(first term) + n(last term)/2)*n.

N1 = 1.
N 100 = N0 + (n-1)*2 = 1 + 198 = 199.

Now sum = ((1+199)/2)*100 = 100*100 = 10,000.

0akshay0 · Answer

Sum of first n positive odd numbers =  n^2

Therefore, Sum of first 100 positive odd numbers =  100^2  = 10000;

Hence option D is correct.

LucienH · Answer

Average of an evenly spaced set: 
(first + last)/2 = (1 + 199)/2 = 100

sum of numbers in the set = n * average = 100 * 100 = 10,000

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What is the sum of the first 100 positive odd numbers?

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