An Arithmetic Sequence has a value, d, called the Common Difference between consecutive terms.
So, we may determine any term using a formula:
Tn = T1 + (n-1)d [where n is the number of the term, starting with 1, and d is the Common Difference]
For example, the sequence 1, 3, 5, 7, 9, 11, ..has T1=1 and d=2, so
T5 = 1 + (5-1)2
T5 = 9
For this problem,
"greater than the term before it" means d is positive
"difference between any 2 consecutive terms is constant" means this is an Arithmetic Sequence
"15th term is 6" means T15 = T1 + (15-1)d = 6 [eq1]
"20th term is 16" means T20 = T1 + (20-1)d = 16 [eq2]
"What's the 300th term?"
T300 = T1 + (300-1)d [So, what's d and T1?]
Subtract eq1 from eq2 to determine d:
T1 + (20-1)d = 16
T1 + (15-1)d = 6
-------------------------
5d = 10
d = 2 [divide both sides by 5]
Then, solve for T1 using either eq1 or eq2:
T1 + (15-1)d = 6
T1 + (14)(2) = 6
T1 = -22
So,
T300 = T1 + (300-1)d
T300 = -22 + 299(2)
T300 = 576
To check, the sequence is:
-22 -20 -18 -16 -14 -12 -10 -8 -6
-4 -2 0 2 4 6 8 10 12 14
16 18 20 22 24 26 28 30 32 34
36 38 40 42 44 46 48 50 52 54
36 38 40 42 44 46 48 50 52 54
56 58 60 62 64 66 68 70 72 74
76 78 80 82 84 86 88 90 92 94
96 98 100 102 104 106 108 110 112 114
116 118 120 122 124 126 128 130 132 134
136 138 140 142 144 146 148 150 152 154
156 158 160 162 164 166 168 170 172 174
76 78 80 82 84 86 88 90 92 94
96 98 100 102 104 106 108 110 112 114
116 118 120 122 124 126 128 130 132 134
136 138 140 142 144 146 148 150 152 154
156 158 160 162 164 166 168 170 172 174
176 178 180 182 184 186 188 190 192 194
196 198 200 202 204 206 208 210 212 214
216 218 220 222 224 226 228 230 232 234
236 238 240 242 244 246 248 250 252 254
256 258 260 262 264 266 268 270 272 274
276 278 280 282 284 286 288 290 292 294
296 298 300 302 304 306 308 310 312 314
316 318 320 322 324 326 328 330 332 334
336 338 340 342 344 346 348 350 352 354
356 358 360 362 364 366 368 370 372 374
196 198 200 202 204 206 208 210 212 214
216 218 220 222 224 226 228 230 232 234
236 238 240 242 244 246 248 250 252 254
256 258 260 262 264 266 268 270 272 274
276 278 280 282 284 286 288 290 292 294
296 298 300 302 304 306 308 310 312 314
316 318 320 322 324 326 328 330 332 334
336 338 340 342 344 346 348 350 352 354
356 358 360 362 364 366 368 370 372 374
376 378 380 382 384 386 388 390 392 394
396 398 400 402 404 406 408 410 412 414
416 418 420 422 424 426 428 430 432 434
436 438 440 442 444 446 448 450 452 454
456 458 460 462 464 466 468 470 472 474
476 478 480 482 484 486 488 490 492 494
496 498 500 502 504 506 508 510 512 514
516 518 520 522 524 526 528 530 532 534
536 538 540 542 544 546 548 550 552 554
556 558 560 562 564 566 568 570 572 574
396 398 400 402 404 406 408 410 412 414
416 418 420 422 424 426 428 430 432 434
436 438 440 442 444 446 448 450 452 454
456 458 460 462 464 466 468 470 472 474
476 478 480 482 484 486 488 490 492 494
496 498 500 502 504 506 508 510 512 514
516 518 520 522 524 526 528 530 532 534
536 538 540 542 544 546 548 550 552 554
556 558 560 562 564 566 568 570 572 574
576