If P is the sum of all the positive odd integers less than 50 and Q is

Question

If P is the sum of all the positive odd integers less than 50 and Q is the sum of all positive even integers less than 50, then P - Q = A. -24 B. 0 C. 1 D. 25 E. 50 image 1.png

Bunuel · Answer

There are 25 odd numbers from 1 to 49, inclusive ((49 - 1)/2 + 1 = 25). Since they represent evenly spaced set their sum = (average) * (number of terms) = (1 + 49)/2 * 25 = 25 *25

There are 24 even numbers from 2 to 48, inclusive ((48 - 2)/2 + 1 = 24) . Since they represent evenly spaced set their sum = (average) * (number of terms) = (2 + 48)/2 * 24 = 25 *24

P - Q = 25 * 25 - 25 *24 = 25 * (25 - 24) = 25.

Answer: D.

Bunuel · Answer

If you know the formulas for the sum of the first positive odd and even integers, you can apply those too.

The sum of the first n positive odd integers = n^2.
Thus, the sum of the first 25 odd integers = 25^2.

The sum of the first n positive even integers = n(n + 1).
Thus, the sum of the first 24 even integers = 24(24 + 1).

Therefore,
P - Q = 25 * 25 - 24 * 25 = 25 * (25 - 24) = 25.

Answer: D.

Bunuel · Answer

Or:

P = 1 + 3 + ... + 47  + 49  (25 terms)
Q = 2 + 4 + ... + 48 (24 terms)

P - Q = (1 - 2) + (3 - 4) + ... + (47 - 48)  + 49  = -1 *24  + 49  = 25.

Answer: D.

sujoykrdatta · Answer

We can easily find the sum of odd numbers and the sum of even numbers and subtract.

Let's look at a different approach:

We are looking at numbers from 1 to 49 (inclusive)

Consider till 48 first: 
There are 24 odd and 24 even numbers, with each odd number 1 less than each corresponding even 
i.e. 
1 is 1 less than 2
3 is 1 less than 4
5 is 1 less than 6
...
47 is 1 less than 48

Thus, the sum of these 24 odd numbers is 24 less than the even numbers

However, the last number is odd and is 49

This increases the sum of odd numbers. Initially the odd numbers were 24 less than the even, but now, after including the last term, the odd numbers are -24 + 49 = 25 greater than the even numbers

Difference = 25

Ans D

MartyMurray · Answer

If  P  is the sum of all the positive odd integers less than 50 and  Q  is the sum of all positive even integers less than 50, then  P - Q  =

Here's a quick approach.

There are 24 even numbers from 2 to 48.

Each of the 24 odd numbers from 3 to 49 is 1 greater than one of the 24 even numbers.

So, the sum of the odd numbers from 3 to 49 is 24 greater than the sum of the even numbers from 2 to 48.

Then, the sum of the odds has to also include 1.

24 + 1 = 25

A. -24
B. 0
C. 1
D. 25
E. 50

Correct answer:   D

GMATinsight · Answer

METHOD_1

P = 1+3+5+....+47+49 = \frac{(First term + Last term)*No. of term }{ 2} = \frac{(1 + 49)*25 }{ 2} = 625

Q = 2+4+6+....+46+48 = \frac{(First term + Last term)*No. of term }{ 2} = \frac{(2 + 48)*24 }{ 2} = 600

P-Q = 625 - 600 = 25

METHOD_2

P = 1+3+5+....+47+49
Q = 2+4+6+....+46+48

P - Q = (1+3+5+7....+47+49) - (2+4+6+....+46+48) = 1+ (3-2) + (5-4) + (7-6) + -------+ (49-48) = 1+(1+1+....24 times) = 1+24 = 25

Answer: Option D

Related Video:
  https://www.youtube.com/watch?v=H81rPGums-g

If P is the sum of all the positive odd integers less than 50 and Q is

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If P is the sum of all the positive odd integers less than 50 and Q is

Prep Toolkit

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