# Power Sum (Text Format)

50 Identities of Power Summation

(09/26/2009 - update)

For each positive integer n,

• 1 + 2 + 3 + ... + n = (1/2)n2 + (1/2)n .

• 12 + 22 + 32 + ... + n2 = (1/3)n3 + (1/2)n2 + (1/6)n .

• 13 + 23 + 33 + ... + n3 = (1/4)n4 + (1/2)n3 + (1/4)n2

• 14 + 24 + 34 + ... + n4 = (1/5)n5 + (1/2)n4 + (1/3)n3 - (1/30)n .

• 15 + 25 + 35 + ... + n5 = (1/6)n6 + (1/2)n5 + (5/12)n4 - (1/12)n2 .

• 16 + 26 + 36 + ... + n6 = (1/7)n7 + (1/2)n6 + (1/2)n5 - (1/6)n3 + (1/42)n .

• 17 + 27 + 37 + ... + n7 = (1/8)n8 + (1/2)n7 + (7/12)n6 - (7/24)n4 + (1/12)n2 .

• 18 + 28 + 38 + ... + n8 = (1/9)n9 + (1/2)n8 + (2/3)n7 - (7/15)n5 + (2/9)n3 - (1/30)n .

• 19 + 29 + 39 + ... + n9 = (1/10)n10 + (1/2)n9 + (3/4)n8 - (7/10)n6 + (1/2)n4 - (3/20)n2 .

• 110 + 210 + 310 + ... + n10 = (1/11)n11 + (1/2)n10 + (5/6)n9 - n7 + n5 - (1/2)n3 + (5/66)n .

• 111 + 211 + 311 + ... + n11 = (1/12)n12 + (1/2)n11 + (11/12)n10 - (11/8)n8 + (11/6)n6 - (11/8)n4 + (5/12)n2 .

• 112 + 212 + 312 + ... + n12 = (1/13)n13 + (1/2)n12 + n11 - (11/6)n9 + (22/7)n7 - (33/10)n5 + (5/3)n3 - (691/2730)n .

• 113 + 213 + 313 + ... + n13 = (1/14)n14 + (1/2)n13 + (13/12)n12 - (143/60)n10 + (143/28)n8 - (143/20)n6 + (65/12)n4 - (691/420)n2 .

• 114 + 214 + 314 + ... + n14 = (1/15)n15 + (1/2)n14 + (7/6)n13 - (91/30)n11 + (143/18)n9 - (143/10)n7 + (91/6)n5 - (691/90)n3 + (7/6)n .

• 115 + 215 + 315 + ... + n15 = (1/16)n16 + (1/2)n15 + (5/4)n14 - (91/24)n12 + (143/12)n10 - (429/16)n8 + (455/12)n6 - (691/24)n4 + (35/4)n2 .

• 116 + 216 + 316 + ... + n16 = (1/17)n17 + (1/2)n16 + (4/3)n15 - (14/3)n13 + (52/3)n11 - (143/3)n9 + (260/3)n7 - (1382/15)n5 + (140/3)n3 - (3617/510)n .

• 117 + 217 + 317 + ... + n17 = (1/18)n18 + (1/2)n17 + (17/12)n16 - (17/3)n14 + (221/9)n12 - (2431/30)n10 + (1105/6)n8 - (11747/45)n6 + (595/3)n4 - (3617/60)n2 .

• 118 + 218 + 318 + ... + n18 = (1/19)n19 + (1/2)n18 + (3/2)n17 - (34/5)n15 + (34)n13 - (663/5)n11 + (1105/3)n9 - (23494/35)n7 + (714)n5 - (3617/10)n3 + (43867/798)n .

• 119 + 219 + 319 + ... + n19 = (1/20)n20 + (1/2)n19 + (19/12)n18 - (323/40)n16 + (323/7)n14 - (4199/20)n12 + (4199/6)n10 - (223193/140)n8 + (2261)n6 - (68723/40)n4 + (43867/84)n2 .

• 120 + 220 + 320 + ... + n20 = (1/21)n21 + (1/2)n20 + (5/3)n19 - (19/2)n17 + (1292/21)n15 - (323)n13 + (41990/33)n11 - (223193/63)n9 + (6460)n7 - (68723/10)n5 + (219335/63)n3 - (174611/330)n .

• 121 + 221 + 321 + ... + n21 = (1/22)n22 + (1/2)n21 + (7/4)n20 - (133/12)n18 + (323/4)n16 - (969/2)n14 + (146965/66)n12 - (223193/30)n10 + (33915/2)n8 - (481061/20)n6 + (219335/12)n4 - (1222277/220)n2 .

• 122 + 222 + 322 + ... + n22 = (1/23)n23 + (1/2)n22 + (11/6)n21 - (77/6)n19 + (209/2)n17 - (3553/5)n15 + (11305/3)n13 - (223193/15)n11 + (124355/3)n9 - (755953/10)n7 + (482537/6)n5 - (1222277/30)n3 + (854513/138)n .

• 123 + 223 + 323 + ... + n23 = (1/24)n24 + (1/2)n23 + (23/12)n22 - (1771/120)n20 + (4807/36)n18 - (81719/80)n16 + (37145/6)n14 - (5133439/180)n12 + (572033/6)n10 - (17386919/80)n8 + (11098351/36)n6 - (28112371/120)n4 + (854513/12)n2 .

• 124 + 224 + 324 + ... + n24 = (1/25)n25 + (1/2)n24 + (2)n23 - (253/15)n21 + (506/3)n19 - (14421/10)n17 + (29716/3)n15 - (10266878/195)n13 + (208012)n11 - (17386919/30)n9 + (22196702/21)n7 - (28112371/25)n5 + (1709026/3)n3 - (236364091/2730)n .

• 125 + 225 + 325 + ... + n25 = (1/26)n26 + (1/2)n25 + (25/12)n24 - (115/6)n22 + (1265/6)n20 - (24035/12)n18 + (185725/12)n16 - (25667195/273)n14 + (1300075/3)n12 - (17386919/12)n10 + (277458775/84)n8 - (28112371/6)n6 + (21362825/6)n4 - (1181820455/1092)n2 .

• 126 + 226 + 326 + ... + n26 = (1/27)n27 + (1/2)n26 + (13/6)n25 - (65/3)n23 + (16445/63)n21 - (16445/6)n19 + (142025/6)n17 - (10266878/63)n15 + (2600150/3)n13 - (20548177/6)n11 + (3606964075/378)n9 - (52208689/3)n7 + (55543345/3)n5 - (1181820455/126)n3 + (8553103/6)n .

• 127 + 227 + 327 + ... + n27 = (1/28)n28 + (1/2)n27 + (9/4)n26 - (195/8)n24 + (4485/14)n22 - (29601/8)n20 + (142025/4)n18 - (15400317/56)n16 + (1671525)n14 - (61644531/8)n12 + (721392815/28)n10 - (469878201/8)n8 + (166630035/2)n6 - (3545461365/56)n4 + (76977927/4)n2 .

• 128 + 228 + 328 + ... + n28 = (1/29)n29 + (1/2)n28 + (7/3)n27 - (273/10)n25 + (390)n23 - (9867/2)n21 + (52325)n19 - (905901/2)n17 + (3120180)n15 - (33193209/2)n13 + (65581165)n11 - (365460823/2)n9 + (333260070)n7 - (709092273/2)n5 + (179615163)n3 - (23749461029/870)n .

• 129 + 229 + 329 + ... + n29 = (1/30)n30 + (1/2)n29 + (29/12)n28 - (609/20)n26 + (1885/4)n24 - (26013/4)n22 + (303485/4)n20 - (8757043/12)n18 + (22621305/4)n16 - (137514723/4)n14 + (1901853785/12)n12 - (10598363867/20)n10 + (4832271015/4)n8 - (6854558639/4)n6 + (5208839727/4)n4 - (23749461029/60)n2 .

• 130 + 230 + 330 + ... + n30 = (1/31)n31 + (1/2)n30 + (5/2)n29 - (203/6)n27 + (1131/2)n25 - (16965/2)n23 + (216775/2)n21 - (2304485/2)n19 + (19959975/2)n17 - (137514723/2)n15 + (731482225/2)n13 - (31795091601/22)n11 + (8053785025/2)n9 - (102818379585/14)n7 + (15626519181/2)n5 - (23749461029/6)n3 + (8615841276005/14322)n .

• 131 + 231 + 331 + ... + n31 = (1/32)n32 + (1/2)n31 + (31/12)n30 - (899/24)n28 + (2697/4)n26 - (175305/16)n24 + (6720025/44)n22 - (14287807/8)n20 + (68751025/4)n18 - (4262956413/32)n16 + (22675948975/28)n14 - (328549279877/88)n12 + (49933467155/4)n10 - (3187369767135/112)n8 + (161474031537/4)n6 - (736233291899/24)n4 + (8615841276005/924)n2 .

• 132 + 232 + 332 + ... + n32 = (1/33)n33 + (1/2)n32 + (8/3)n31 - (124/3)n29 + (7192/9)n27 - (70122/5)n25 + (2337400/11)n23 - (57151228/21)n21 + (28947800)n19 - (4262956413/17)n17 + (36281518360/21)n15 - (101092086116/11)n13 + (36315248840)n11 - (2124913178090/21)n9 + (184541750328)n7 - (2944933167596/15)n5 + (68926730208040/693)n3 - (7709321041217/510)n .

• 133 + 233 + 333 + ... + n33 = (1/34)n34 + (1/2)n33 + (11/4)n32 - (682/15)n30 + (19778/21)n28 - (89001/5)n26 + (292175)n24 - (28575614/7)n22 + (47763870)n20 - (15630840181/34)n18 + (49887087745/14)n16 - (21662589882)n14 + (99866934310)n12 - (2337404495899/7)n10 + (761234720103)n8 - (16197132421778/15)n6 + (17231682552010/21)n4 - (84802531453387/340)n2 .

• 134 + 234 + 334 + ... + n34 = (1/35)n35 + (1/2)n34 + (17/6)n33 - (748/15)n31 + (23188/21)n29 - (336226/15)n27 + (397358)n25 - (42242212/7)n23 + (77331980)n21 - (822675799)n19 + (49887087745/7)n17 - (245509351996/5)n15 + (261190443580)n13 - (7224704805506/7)n11 + (2875775609278)n9 - (78671786048636/15)n7 + (117175441353668/21)n5 - (84802531453387/30)n3 + (2577687858367/6)n .

• 135 + 235 + 335 + ... + n35 = (1/36)n36 + (1/2)n35 + (35/12)n34 - (1309/24)n32 + (11594/9)n30 - (168113/6)n28 + (534905)n26 - (52802765/6)n24 + (123028150)n22 - (5758730593/4)n20 + (249435438725/18)n18 - (429641365993/4)n16 + (652976108950)n14 - (18061762013765/6)n12 + (10065214632473)n10 - (137675625585113/6)n8 + (292938603384170/9)n6 - (593617720173709/24)n4 + (90219075042845/12)n2 .

• 136 + 236 + 336 + ... + n36 = (1/37)n37 + (1/2)n36 + (3)n35 - (119/2)n33 + (1496)n31 - (34782)n29 + (2139620/3)n27 - (63363318/5)n25 + (192565800)n23 - (2468027397)n21 + (498870877450/19)n19 - (227457193761)n17 + (1567142661480)n15 - (108370572082590/13)n13 + (32940702433548)n11 - (275351251170226/3)n9 + (1171754413536680/7)n7 - (1780853160521127/10)n5 + (90219075042845)n3 - (26315271553053477373/1919190)n .

• 137 + 237 + 337 + ... + n37 = (1/38)n38 + (1/2)n37 + (37/12)n36 - (259/4)n34 + (6919/4)n32 - (214489/5)n30 + (2827355/3)n28 - (1172221383/65)n26 + (296872275)n24 - (8301546699/2)n22 + (1845822246565/38)n20 - (935101796573/2)n18 + (7248034809345/2)n16 - (2004855583527915/91)n14 + (101567165836773)n12 - (5093998146649181/15)n10 + (5419364162607145/7)n8 - (21963855646427233/20)n6 + (3338105776585265/4)n4 - (26315271553053477373/103740)n2 .

• 138 + 238 + 338 + ... + n38 = (1/39)n39 + (1/2)n38 + (19/6)n37 - (703/10)n35 + (11951/6)n33 - (262922/5)n31 + (3704810/3)n29 - (14848137518/585)n27 + (451245858)n25 - (6857799447)n23 + (1845822246565/21)n21 - (935101796573)n19 + (8100744786915)n17 - (5078967478270718/91)n15 + (296888638599798)n13 - (17597448142969898/15)n11 + (205935838179071510/63)n9 - (59616179611731061/10)n7 + (12684801951024007/2)n5 - (26315271553053477373/8190)n3 + (2929993913841559/6)n .

• 139 + 239 + 339 + ... + n39 = (1/40)n40 + (1/2)n39 + (13/4)n38 - (9139/120)n36 + (9139/4)n34 - (5126979/80)n32 + (4816253/3)n30 - (7424068759/210)n28 + (676868787)n26 - (89151392811/8)n24 + (2181426291395/14)n22 - (36468970066347/20)n20 + (35103227409965/2)n18 - (7618451217406077/56)n16 + (827046921813723)n14 - (114383412929304337/30)n12 + (267716589632792963/21)n10 - (2325031004857511379/80)n8 + (164902425363312091/4)n6 - (26315271553053477373/840)n4 + (38089920879940267/4)n2 .

• 140 + 240 + 340 + ... + n40 = (1/41)n41 + (1/2)n40 + (10/3)n39 - (247/3)n37 + (18278/7)n35 - (155363/2)n33 + (6214520/3)n31 - (1024009484/21)n29 + (3008305720/3)n27 - (89151392811/5)n25 + (1896892427300/7)n23 - (3473235244414)n21 + (36950765694700)n19 - (2240720946295905/7)n17 + (2205458458169928)n15 - (35194896285939796/3)n13 + (10708663585311718520/231)n11 - (775010334952503793/6)n9 + (235574893376160130)n7 - (26315271553053477373/105)n5 + (380899208799402670/3)n3 - (261082718496449122051/13530)n .

• 141 + 241 + 341 + ... + n41 = (1/42)n42 + (1/2)n41 + (41/12)n40 - (533/6)n38 + (374699/126)n36 - (374699/4)n34 + (31849415/12)n32 - (20992194422/315)n30 + (4405019090/3)n28 - (281169777327/10)n26 + (19443147379825/42)n24 - (71201322510487/11)n22 + (75749069674135)n20 - (30623186266044035/42)n18 + (11302974598120881/2)n16 - (103070767694537974/3)n14 + (109763801749445114830/693)n12 - (31775423733052655513/60)n10 + (4829285314211282665/4)n8 - (1078926133675192572293/630)n6 + (7808433780387754735/6)n4 - (261082718496449122051/660)n2 .

• 142 + 242 + 342 + ... + n42 = (1/43)n43 + (1/2)n42 + (7/2)n41 - (287/3)n39 + (10127/3)n37 - (1124097/10)n35 + (222945905/66)n33 - (1354335124/15)n31 + (2126560940)n29 - (656062813763/15)n27 + (777725895193)n25 - (130019806323498/11)n23 + (151498139348270)n21 - (1611746645581265)n19 + (13962498032972853)n17 - (1442990747723531636/15)n15 + (16886738730683863820/33)n13 - (20220724193760780781/10)n11 + (33804997199478978655/6)n9 - (1078926133675192572293/105)n7 + (10931807292542856629)n5 - (1827579029475143854357/330)n3 + (1520097643918070802691/1806)n .

• 143 + 243 + 343 + ... + n43 = (1/44)n44 + (1/2)n43 + (43/12)n42 - (12341/120)n40 + (22919/6)n38 - (16112057/120)n36 + (563921995/132)n34 - (14559102583/120)n32 + (9144212042/3)n30 - (4030100141687/60)n28 + (2572477961023/2)n26 - (931808611985069/44)n24 + (296109999635255)n22 - (13861021151998879/4)n20 + (66709712824203631/2)n18 - (15512150538027965087/60)n16 + (363064882709703072130/231)n14 - (869491140331713573583/120)n12 + (290722975915519216433/12)n10 - (46393823748033280608599/840)n8 + (470067713579342835047/6)n6 - (78585898267431185737351/1320)n4 + (1520097643918070802691/84)n2 .

• 144 + 244 + 344 + ... + n44 = (1/45)n45 + (1/2)n44 + (11/3)n43 - (3311/30)n41 + (38786/9)n39 - (4790071/30)n37 + (16112057/3)n35 - (14559102583/90)n33 + (12978881608/3)n31 - (1528658674433/15)n29 + (56594515142506/27)n27 - (931808611985069/25)n25 + (13028839983951220/23)n23 - (152471232671987669/21)n21 + (77242825375393678)n19 - (10037273877547506821/15)n17 + (290451906167762457704/63)n15 - (735723272588373023801/30)n13 + (290722975915519216433/3)n11 - (510332061228366086694589/1890)n9 + (1477355671249363195862/3)n7 - (78585898267431185737351/150)n5 + (16721074083098778829601/63)n3 - (27833269579301024235023/690)n .

• 145 + 245 + 345 + ... + n45 = (1/46)n46 + (1/2)n45 + (15/4)n44 - (473/4)n42 + (19393/4)n40 - (756327/4)n38 + (80560285/12)n36 - (856417799/4)n34 + (24335403015/4)n32 - (1528658674433/10)n30 + (141486287856265/42)n28 - (645098269835817/10)n26 + (48858149939817075/46)n24 - (207915317279983185/14)n22 + (347592714189271551/2)n20 - (10037273877547506821/6)n18 + (181532441354851536065/14)n16 - (315309973966445581629/4)n14 + (1453614879577596082165/4)n12 - (510332061228366086694589/420)n10 + (11080167534370223968965/4)n8 - (78585898267431185737351/20)n6 + (83605370415493894148005/28)n4 - (83499808737903072705069/92)n2 .

• 146 + 246 + 346 + ... + n46 = (1/47)n47 + (1/2)n46 + (23/6)n45 - (253/2)n43 + (10879/2)n41 - (446039/2)n39 + (50078015/6)n37 - (19697609377/70)n35 + (16961038465/2)n33 - (1134166113289/5)n31 + (112213262782555/21)n29 - (1648584467358199/15)n27 + (1954325997592683)n25 - (207915317279983185/7)n23 + (380696782207297413)n21 - (12150384167557508257/3)n19 + (245602714774210901735/7)n17 - (2417376467076082792489/10)n15 + (2571780171560362299215/2)n13 - (1067057946204765453997777/210)n11 + (84947951096838383762065/6)n9 - (258210808592988181708439/10)n7 + (384584703911271913080823/14)n5 - (27833269579301024235023/2)n3 + (596451111593912163277961/282)n .

• 147 + 247 + 347 + ... + n47 = (1/48)n48 + (1/2)n47 + (47/12)n46 - (1081/8)n44 + (511313/84)n42 - (20963833/80)n40 + (123877195/12)n38 - (925787640719/2520)n36 + (46892282815/4)n34 - (53305807324583/160)n32 + (1054804670156017/126)n30 - (11069067137976479/60)n28 + (7065640145142777/2)n26 - (3257339970719736565/56)n24 + (1626613523976634401/2)n22 - (571068055875202888079/60)n20 + (11543327594387912381545/126)n18 - (113616693952575891246983/160)n16 + (17267666866191004009015/4)n14 - (50151723471623976337895519/2520)n12 + (798510740310280807363411/12)n10 - (12135908003870444540296633/80)n8 + (18075481083829779914798681/84)n6 - (1308163670227148139046081/8)n4 + (596451111593912163277961/12)n2 .

• 148 + 248 + 348 + ... + n48 = (1/49)n49 + (1/2)n48 + (4)n47 - (2162/15)n45 + (47564/7)n43 - (1533939/5)n41 + (38116060/3)n39 - (50042575174/105)n37 + (16077354108)n35 - (4845982484053/10)n33 + (272207656814456/21)n31 - (1526767881100204/5)n29 + (18841707053714072/3)n27 - (3908807964863683878/35)n25 + (1697335851106053288)n23 - (326324603357258793188/15)n21 + (4860348460794910476440/21)n19 - (340850081857727673740949/170)n17 + (13814133492952803207212)n15 - (100303446943247952675791038/1365)n13 + (290367541931011202677604)n11 - (12135908003870444540296633/15)n9 + (72301924335319119659194724/49)n7 - (7848982021362888834276486/5)n5 + (2385804446375648653111844/3)n3 - (5609403368997817686249127547/46410)n .

• 149 + 249 + 349 + ... + n49 = (1/50)n50 + (1/2)n49 + (49/12)n48 - (2303/15)n46 + (7567)n44 - (3579191/10)n42 + (93384347/6)n40 - (9218369111/15)n38 + (65649195941/3)n36 - (237453141718597/340)n34 + (238181699712649/12)n32 - (37405813086954998/75)n30 + (32972987343999626/3)n28 - (13680827877022893573/65)n26 + (3465394029341525463)n24 - (726813889295712766646/15)n22 + (1701121961278218666754/3)n20 - (5567218003676218671102167/1020)n18 + (169223135288671839288347/4)n16 - (50151723471623976337895519/195)n14 + (3557002388654887232800649/3)n12 - (594659492189651782474535017/150)n10 + (18075481083829779914798681/2)n8 - (64100019841130258813257969/5)n6 + (29226104468101696000620089/3)n4 - (39265823582984723803743892829/13260)n2 .

• 150 + 250 + 350 + ... + n50 = (1/51)n51 + (1/2)n50 + (25/6)n49 - (490/3)n47 + (75670/9)n45 - (416185)n43 + (56941675/3)n41 - (92183691110/117)n39 + (88715129650/3)n37 - (33921877388371/34)n35 + (541322044801475/18)n33 - (2413278263674516/3)n31 + (56849978179309700/3)n29 - (45602759590076311910/117)n27 + (6930788058683050926)n25 - (316006038824222942020/3)n23 + (12150871151987276191100/9)n21 - (1465057369388478597658465/102)n19 + (248857551895105646012275/2)n17 - (100303446943247952675791038/117)n15 + (13680778417903412433848650/3)n13 - (594659492189651782474535017/33)n11 + (451887027095744497869967025/9)n9 - (91571456915900369733225670)n7 + (292261044681016960006200890/3)n5 - (196329117914923619018719464145/3978)n3 + (495057205241079648212477525/66)n.

(09/26/2009 - Update)

In-Text or Website Citation
Tue N. Vu, 50 Identities of Power Summation, 09/26/2009 (update), from Series Math Study Resource.

We all agree that your theory is crazy, but is it crazy enough?
Niels Bohr (1885-1962)