I found a trick or a shortcut to find the sum of the first half of consecutive integers( STRAIGHT,EVEN AND ODD) given the sum of one half of the set. This works only when the total number of elements in the set is even. but definitely saves a few seconds.

Let me illustrate with examples:

1.

CONSECUTIVE INTEGERS : 6 TO 15 The Sum of the greater 5 numbers in a set of 10 consecutive integers is 65. Find the sum of the first 5 numbers.

Short cut:

Step 1: Multiply the no.of elements in each half : in this case 5 each and the spacing between each number in the set. in this case 1. ie., 5*5*1 = 25

Step 2: If the given sum is that of the greater numbers in the set, then subtract '25' to get the sum of the lower 5 numbers, i.e, 65-25 = 40 is the Answer

or if the given sum is that of the lower 5 consecutive numbers, then add '25' to get the sum of the greater 5 numbers i.e., 40+25 = 65.

2.

Lets try this with 6 consecutive EVEN integers:Find the sum of lower half of the numbers in a set of 6 consecutive even integers if the sum of the latter half is 30.

Step 1: 3*3*2 ( remember each half has 3 elements and the spacing between the elements is 2 as they are even) = 18

step 2: the given sum - step 1= 30 - 18 = 12 ( the sum of the even integers 2,4 and 6) is the answer.

3

.Lets try for the 16 consecutive integers from 8 to 23. given sum of the greater 8 numbers in the set = 156step 1: 8*8*1=64

step 2: given sum - step 1 = 156-64= 92 (which is the sum of numbers starting 8 thru 15)

4. Now lets try 24 consecutive ODD integers:

Find the sum of the second half of the elements of a set when the first half sums up to 168. The set contains Consecutive Odd integers.

Step 1: 12*12*2 = 288 ( Odd numbers are spaced evenly)

Step 2: given sum + 168 = 288+168 = 456

Check it out : the numbers are 3 to 49 inclusive.

try more examples. But remember it works only on CONSECUTIVE INTEGERS WITH EVEN NUMBER OF ELEMENTS. And when the sum of lower half is given, you need to ADD the given sum to step 1 and when the sum of greater half is given u need to SUBTRACT step 1 from the given sum.

Hope this helps!