Property 5 - Square Numbers

Property 1	Property 2	Property 3	Property 4	Property 5
Property 6	Property 7	Property 8

Before you understand this property, you are adviced to read: What are Natural Numbers ? What are Square Numbers ? Observe the following Table 1: Odd Square Number Sum of Two Consecutive Natural Number 9 4 + 5 25 12 + 13 49 24 + 25 81 40 + 41 121 60 + 61 169 84 + 85 225 112 + 113 289 144 + 145 361 180 + 181 441 220 + 221 529 264 + 265 In the above table 1: Column 1 represents Odd Square numbers Column 2 represents Sum of Consecutive Natural Numbers Here, you can observe that in each row sum of consecutive natural numbers is equal to its corresponding odd square number. (Note - This property does not applied to even square numbers like 4, 16, 36, 64 …. etc ) Now, learn the following two formulas to find these two consecutive natural numbers First Consecutive Natural Number = (Square Number - 1) / 2 Second Consecutive Natural Number = (Square Number + 1) / 2 Use these formulas to solve following examples: Example 1: Express 49 as the sum of two consecutive natural number Solution: Given square numbers = 49 Let's find first consecutive natural and apply the above mentioned formula First Consecutive Natural Number = (Square Number - 1) / 2 Put value of Square number and we get: = (49 - 1) / 2 Solve bracket and we get: = 48 / 2 Solve division expression and we get: = 24 So, First Consecutive Natural Number = 24 Now find second consecutive natural and apply the above mentioned formula Second Consecutive Natural Number = (Square Number + 1) / 2 Put value of Square number and we get: = (49 + 1) / 2 Solve bracket and we get: = 50 / 2 Solve division expression and we get: = 25 So, Second Consecutive Natural Number = 24 Hence, 49 can be expressed as the sum of 24 and 25 (you can cross-check the answer from the above table 1) Example 2: Express 361 as the sum of two consecutive integers Solution: Given Odd square numbers = 361 Let's find first consecutive natural and apply the above mentioned formula First Consecutive Natural Number = (Square Number - 1) / 2 Put value of Square number and we get: = (361 - 1) / 2 Solve bracket and we get: = 360 / 2 Solve division expression and we get: = 180 So, First Consecutive Natural Number = 180 Now find second consecutive natural and apply the above mentioned formula Second Consecutive Natural Number = (Square Number + 1) / 2 Put value of Square number and we get: = (361 + 1) / 2 Solve bracket and we get: = 362 / 2 Solve division expression and we get: = 181 So, Second Consecutive Natural Number = 181 Hence, 361 can be expressed as the sum of 180 and 181 (you can cross-check the answer from the above table 1) Example 3: Express 232 as the sum of two consecutive numbers Solution: Convert the given numbers (232) into square number and we get: Square Number = 529 Let's find first consecutive natural and apply the above mentioned formula First Consecutive Natural Number = (Square Number - 1) / 2 Put value of Square number and we get: = (529 - 1) / 2 Solve bracket and we get: = 528 / 2 Solve division expression and we get: = 264 So, First Consecutive Natural Number = 264 Now find second consecutive natural and apply the above mentioned formula Second Consecutive Natural Number = (Square Number + 1) / 2 Put value of Square number and we get: = (529 + 1) / 2 Solve bracket and we get: = 530 / 2 Solve division expression and we get: = 265 So, Second Consecutive Natural Number = 265 Hence, 361 can be expressed as the sum of 180 and 181 (you can cross-check the answer from the above table 1)