sagnik2422 wrote:

12 + 13 + 14 + ... 51 + 52 + 53 = ?

A. 1361

B. 1362

C. 1363

D. 1364

E. 1365

explanation showed adding in pairs which added to 65 and then y - x + 1 = 53 - 12 + 1 = 42

and then 21 pairs add to 65 and 65 x 21 and looked at the last digit which was 5

MY QUESTION : WHY IS IT 21 PAIRS? How would we know this?

You can do this question in different ways:

Sum = 12 + 13 + 14 + ... 51 + 52 + 53

Method 1:

Sum of n consecutive positive integers starting from 1 is given as n(n+1)/2

Sum of first 53 positive integers = 53*54/2

Sum of first 11 positive integers = 11*12/2

Sum = 12 + 13 + 14 + ... 51 + 52 + 53 = 53*54/2 - 11*12/2 = 53*27 - 66

Note that 3*7 is 21 so 53*27 will end with 1 and when we subtract 66 from it, last digit will be 5. So answer must be (E)

Method 2:

From 12 to 53, how many numbers are there? 53 - 12 + 1 = 42 numbers

Number of positive integers from A to B are calculated as B - A + 1. You add an extra 1 to make up for the 12 that you subtracted. Since you want to include 12 as well in your counting, you add 1.

The average of these terms is (First term + last term)/2 = (53 + 12)/2 = 32.5

Average of a list is the number which can replace all numbers such that the sum of the list stays the same. So sum of the list can be calculated as Average*Number of elements in the list

Sum = 32.5*42 = 65*21

Ends with 5 so answer (E)