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

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 = 2²
Sum of the first 3 odd numbers = 3²
Sum of the first 4 odd numbers = 4²
Sum of the first 5 odd numbers = 5²
In general, the sum of the first n odd numbers = n²

So, the sum of the first 100 odd numbers = 100² = 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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==> Since 1+3+5+….+(2n-1)=n 2, 1+3+…….(2*100-1)=100 2 =10,000 is derived.

Hence, the answer is D.

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.

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.

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