Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."
Part A. Write down the positive integers, cross out every second, and form the partial sums of the remaining.
1 2 3 4 5 6 7 8 9 10 11
1 4 9 16 25 36
Does the pattern of squares continue?
Part B. As before, but cross out every third, form partial sums, then cross out every second and for a second partial sums.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 7 12 19 27 37 48 61 75 91
1 8 27 64 125 216
Does the pattern of cubes continue?
Part A:
I'd expect the patterns to continue, based on what has been called the calculus of finite differences, but worked backwards, analogous to integration, rather than differentiation. Once the alternate numbers have been found to be the appropriate basis for the integration, such as 1,3,5,... (the odd numbers), they will continue to work, as, going in the other direction, their differences are a constant 2.
In each of the pairs of numbers below, the first member is the partial sum and the second is the square root, all of which are integers
1 1 4 2 9 3 16 4 25 5 36 6 49 7 64 8 81 9 100 10 121 11 144 12 169 13 196 14 225 15 256 16 289 17 324 18 361 19 400 20 441 21 484 22 529 23 576 24 625 25 676 26 729 27 784 28 841 29 900 30 961 31 1024 32 1089 33 1156 34 1225 35 1296 36 1369 37 1444 38 1521 39 1600 40 1681 41 1764 42 1849 43 1936 44 2025 45 2116 46 2209 47 2304 48 2401 49 2500 50 2601 51 2704 52 2809 53 2916 54 3025 55 3136 56 3249 57 3364 58 3481 59 3600 60 3721 61 3844 62 3969 63 4096 64 4225 65 4356 66 4489 67 4624 68 4761 69 4900 70 5041 71 5184 72 5329 73 5476 74 5625 75 5776 76 5929 77 6084 78 6241 79 6400 80 6561 81 6724 82 6889 83 7056 84 7225 85 7396 86 7569 87 7744 88 7921 89 8100 90 8281 91 8464 92 8649 93 8836 94 9025 95 9216 96 9409 97 9604 98 9801 99 10000 100 10201 101 10404 102 10609 103 10816 104 11025 105 11236 106 11449 107 11664 108 11881 109 12100 110 12321 111 12544 112 12769 113 12996 114 13225 115 13456 116 13689 117 13924 118 14161 119 14400 120 14641 121 14884 122 15129 123 15376 124 15625 125 15876 126 16129 127 16384 128 16641 129 16900 130 17161 131 17424 132 17689 133 17956 134 18225 135 18496 136 18769 137 19044 138 19321 139 19600 140 19881 141 20164 142 20449 143 20736 144 21025 145 21316 146 21609 147 21904 148 22201 149 22500 150 22801 151 23104 152 23409 153 23716 154 24025 155 24336 156 24649 157 24964 158 25281 159 25600 160 25921 161 26244 162 26569 163 26896 164 27225 165 27556 166 27889 167 28224 168 28561 169 28900 170 29241 171 29584 172 29929 173 30276 174 30625 175 30976 176 31329 177 31684 178 32041 179 32400 180 32761 181 33124 182 33489 183 33856 184 34225 185 34596 186 34969 187 35344 188 35721 189 36100 190 36481 191 36864 192 37249 193 37636 194 38025 195 38416 196 38809 197 39204 198 39601 199 40000 200 40401 201 40804 202 41209 203 41616 204 42025 205 42436 206 42849 207 43264 208 43681 209 44100 210 44521 211 44944 212 45369 213 45796 214 46225 215 46656 216 47089 217 47524 218 47961 219 48400 220 48841 221 49284 222 49729 223 50176 224 50625 225 51076 226 51529 227 51984 228 52441 229 52900 230 53361 231 53824 232 54289 233 54756 234 55225 235 55696 236 56169 237 56644 238 57121 239 57600 240 58081 241 58564 242 59049 243 59536 244 60025 245 60516 246 61009 247 61504 248 62001 249 62500 250 63001 251 63504 252 64009 253 64516 254 65025 255 65536 256 66049 257 66564 258 67081 259 67600 260 68121 261 68644 262 69169 263 69696 264 70225 265 70756 266 71289 267 71824 268 72361 269 72900 270 73441 271 73984 272 74529 273 75076 274 75625 275 76176 276 76729 277 77284 278 77841 279 78400 280 78961 281 79524 282 80089 283 80656 284 81225 285 81796 286 82369 287 82944 288 83521 289 84100 290 84681 291 85264 292 85849 293 86436 294 87025 295 87616 296 88209 297 88804 298 89401 299 90000 300 90601 301 91204 302 91809 303 92416 304 93025 305 93636 306 94249 307 94864 308 95481 309 96100 310 96721 311 97344 312 97969 313 98596 314 99225 315 99856 316 100489 317 101124 318 101761 319 102400 320 103041 321 103684 322 104329 323 104976 324 105625 325 106276 326 106929 327 107584 328 108241 329 108900 330 109561 331 110224 332 110889 333 111556 334 112225 335 112896 336 113569 337 114244 338 114921 339 115600 340 116281 341 116964 342 117649 343 118336 344 119025 345 119716 346 120409 347 121104 348 121801 349 122500 350 123201 351 123904 352 124609 353 125316 354 126025 355 126736 356 127449 357 128164 358 128881 359 129600 360 130321 361 131044 362 131769 363 132496 364 133225 365 133956 366 134689 367 135424 368 136161 369 136900 370 137641 371 138384 372 139129 373 139876 374 140625 375 141376 376 142129 377 142884 378 143641 379 144400 380 145161 381 145924 382 146689 383 147456 384 148225 385 148996 386 149769 387 150544 388 151321 389 152100 390 152881 391 153664 392 154449 393 155236 394 156025 395 156816 396 157609 397 158404 398 159201 399 160000 400 160801 401 161604 402 162409 403 163216 404 164025 405 164836 406 165649 407 166464 408 167281 409 168100 410 168921 411 169744 412 170569 413 171396 414 172225 415 173056 416 173889 417 174724 418 175561 419 176400 420 177241 421 178084 422 178929 423 179776 424 180625 425 181476 426 182329 427 183184 428 184041 429 184900 430 185761 431 186624 432 187489 433 188356 434 189225 435 190096 436 190969 437 191844 438 192721 439 193600 440 194481 441 195364 442 196249 443 197136 444 198025 445 198916 446 199809 447 200704 448 201601 449 202500 450 203401 451 204304 452 205209 453 206116 454 207025 455 207936 456 208849 457 209764 458 210681 459 211600 460 212521 461 213444 462 214369 463 215296 464 216225 465 217156 466 218089 467 219024 468 219961 469 220900 470 221841 471 222784 472 223729 473 224676 474 225625 475 226576 476 227529 477 228484 478 229441 479 230400 480 231361 481 232324 482 233289 483 234256 484 235225 485 236196 486 237169 487 238144 488 239121 489 240100 490 241081 491 242064 492 243049 493 244036 494 245025 495 246016 496 247009 497 248004 498 249001 499 250000 500
Part B:
This is more complicated, so it's harder to predict. I'd just guess it was similar based on the sample shown in the puzzle, through 216.
The pairs below show the partial sums with their cube roots, again all still integers.
1 1 8 2 27 3 64 4 125 5 216 6 343 7 512 8 729 9 1000 10 1331 11 1728 12 2197 13 2744 14 3375 15 4096 16 4913 17 5832 18 6859 19 8000 20 9261 21 10648 22 12167 23 13824 24 15625 25 17576 26 19683 27 21952 28 24389 29 27000 30 29791 31 32768 32 35937 33 39304 34 42875 35 46656 36 50653 37 54872 38 59319 39 64000 40 68921 41 74088 42 79507 43 85184 44 91125 45 97336 46 103823 47 110592 48 117649 49 125000 50 132651 51 140608 52 148877 53 157464 54 166375 55 175616 56 185193 57 195112 58 205379 59 216000 60 226981 61 238328 62 250047 63 262144 64 274625 65 287496 66 300763 67 314432 68 328509 69 343000 70 357911 71 373248 72 389017 73 405224 74 421875 75 438976 76 456533 77 474552 78 493039 79 512000 80 531441 81 551368 82 571787 83 592704 84 614125 85 636056 86 658503 87 681472 88 704969 89 729000 90 753571 91 778688 92 804357 93 830584 94 857375 95 884736 96 912673 97 941192 98 970299 99 1000000 100 1030301 101 1061208 102 1092727 103 1124864 104 1157625 105 1191016 106 1225043 107 1259712 108 1295029 109 1331000 110 1367631 111 1404928 112 1442897 113 1481544 114 1520875 115 1560896 116 1601613 117 1643032 118 1685159 119 1728000 120 1771561 121 1815848 122 1860867 123 1906624 124 1953125 125 2000376 126 2048383 127 2097152 128 2146689 129 2197000 130 2248091 131 2299968 132 2352637 133 2406104 134 2460375 135 2515456 136 2571353 137 2628072 138 2685619 139 2744000 140 2803221 141 2863288 142 2924207 143 2985984 144 3048625 145 3112136 146 3176523 147 3241792 148 3307949 149 3375000 150 3442951 151 3511808 152 3581577 153 3652264 154 3723875 155 3796416 156 3869893 157 3944312 158 4019679 159 4096000 160 4173281 161 4251528 162 4330747 163 4410944 164 4492125 165 4574296 166 4657463 167 4741632 168 4826809 169 4913000 170 5000211 171 5088448 172 5177717 173 5268024 174 5359375 175 5451776 176 5545233 177 5639752 178 5735339 179 5832000 180 5929741 181 6028568 182 6128487 183 6229504 184 6331625 185 6434856 186 6539203 187 6644672 188 6751269 189 6859000 190 6967871 191 7077888 192 7189057 193 7301384 194 7414875 195 7529536 196 7645373 197 7762392 198 7880599 199 8000000 200 8120601 201 8242408 202 8365427 203 8489664 204 8615125 205 8741816 206 8869743 207 8998912 208 9129329 209 9261000 210 9393931 211 9528128 212 9663597 213 9800344 214 9938375 215 10077696 216 10218313 217 10360232 218 10503459 219 10648000 220 10793861 221 10941048 222 11089567 223 11239424 224 11390625 225 11543176 226 11697083 227 11852352 228 12008989 229 12167000 230 12326391 231 12487168 232 12649337 233 12812904 234 12977875 235 13144256 236 13312053 237 13481272 238 13651919 239 13824000 240 13997521 241 14172488 242 14348907 243 14526784 244 14706125 245 14886936 246 15069223 247 15252992 248 15438249 249 15625000 250
DefDbl A-Z
Dim crlf$, sums(3, 10000)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
lim = 1000
For i = 1 To lim
sums(0, i) = i
Next
For i = 1 To lim / 2
sums(1, i) = sums(1, i - 1) + sums(0, 2 * i - 1)
Text1.Text = Text1.Text & Str(sums(1, i)) & Str(Sqr(sums(1, i))) & " "
DoEvents
Next
Text1.Text = Text1.Text & crlf & crlf
lim = 1000
For i = 1 To lim
sums(0, i) = i
Next
src = 1
For i = 1 To lim / 2
sums(1, i) = sums(1, i - 1) + sums(0, src)
src = src + 1
If src Mod 3 = 0 Then src = src + 1
DoEvents
Next
For i = 1 To lim / 4
sums(2, i) = sums(2, i - 1) + sums(1, 2 * i - 1)
Text1.Text = Text1.Text & Str(sums(2, i)) & Str((sums(2, i)) ^ (1 / 3)) & " "
DoEvents
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2017-09-25 12:32:44 |
answers
