We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
78
2
avatar

Determine the value of \(-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 10000\), where the signs change after each perfect square.

 May 3, 2019
 #1
avatar
+1

This reduces to the sum of alternating(+or-) cubes from 1 to the square root of 10,000 or 100 terms as follows:

 

n=0;a= (-n^3)+(n+1)^3;printa,", ",;n++;if(n<100, goto1, discard=0;

sum(1 , 7 , 19 , 37 , 61 , 91 , 127 , 169 , 217 , 271 , 331 , 397 , 469 , 547 , 631 , 721 , 817 , 919 , 1027 , 1141 , 1261 , 1387 , 1519 , 1657 , 1801 , 1951 , 2107 , 2269 , 2437 , 2611 , 2791 , 2977 , 3169 , 3367 , 3571 , 3781 , 3997 , 4219 , 4447 , 4681 , 4921 , 5167 , 5419 , 5677 , 5941 , 6211 , 6487 , 6769 , 7057 , 7351 , 7651 , 7957 , 8269 , 8587 , 8911 , 9241 , 9577 , 9919 , 10267 , 10621 , 10981 , 11347 , 11719 , 12097 , 12481 , 12871 , 13267 , 13669 , 14077 , 14491 , 14911 , 15337 , 15769 , 16207 , 16651 , 17101 , 17557 , 18019 , 18487 , 18961 , 19441 , 19927 , 20419 , 20917 , 21421 , 21931 , 22447 , 22969 , 23497 , 24031 , 24571 , 25117 , 25669 , 26227 , 26791 , 27361 , 27937 , 28519 , 29107 , 29701) 

=1,000,000 - which is the sum of the sequence.

 May 3, 2019
 #2
avatar
0

Another simple way:

 

ALL the pluses and minues cancel each other out as follows:

-1 + 2 +3 + 4 - 5 -6 -7 -8 -9 +10 +11 + 12 + 13 + 14 + 15 + 16 - 17 - 18.........etc =64 - which is the 4^3 and the LAST cube in all these additions and subtractions. The process holds true for ALL 99 terms, leaving the value of the LAST cube, which is 100. Therefore:

100^3 =1,000,000 - which is the sum of this sequence.

 May 4, 2019

5 Online Users