Complete the pattern for the given sequences: Sequence D (red), Sequence C (blue), Sequence D (green), and sequence a).

Here are the solutions to the patterns and the word problem.

Train Car Sequences

• Sequence D (red cars): 538, 547, 556, \text{___}, \text{___}, \text{___}

  • Pattern: Add 9 to the previous number ($547 - 538 = 9$, $556 - 547 = 9$).
  • Next numbers: $$556 + 9 = 565$$ $$565 + 9 = 574$$ $$574 + 9 = 583$$
  • The completed sequence is: $538, 547, 556, \boxed{565}, \boxed{574}, \boxed{583}$.

• Sequence C (blue cars): 729, 629, 529, \text{___}, \text{___}, \text{___}

  • Pattern: Subtract 100 from the previous number ($629 - 729 = -100$, $529 - 629 = -100$).
  • Next numbers: $$529 - 100 = 429$$ $$429 - 100 = 329$$ $$329 - 100 = 229$$
  • The completed sequence is: $729, 629, 529, \boxed{429}, \boxed{329}, \boxed{229}$.

• Sequence D (green cars): 347, 354, 361, \text{___}, \text{___}, \text{___}

  • Pattern: Add 7 to the previous number ($354 - 347 = 7$, $361 - 354 = 7$).
  • Next numbers: $$361 + 7 = 368$$ $$368 + 7 = 375$$ $$375 + 7 = 382$$
  • The completed sequence is: $347, 354, 361, \boxed{368}, \boxed{375}, \boxed{382}$.

Complete the pattern.

• a) 373, 374, 376, \text{___}, \text{___}, \text{___}

  • Pattern: Add 1, then add 2, then add 3, and so on. (The amount added increases by 1 each time).
  • Next numbers: $$376 + 3 = 379$$ $$379 + 4 = 383$$ $$383 + 5 = 388$$
  • The completed sequence is: $373, 374, 376, \boxed{379}, \boxed{383}, \boxed{388}$.

• b) 650, 653, 659, \text{___}, \text{___}, \text{___}

  • Pattern: Add 3, then add 6, then add 12, and so on. (The amount added doubles each time).
  • Next numbers: $$659 + 12 = 671$$ $$671 + 24 = 695$$ $$695 + 48 = 743$$
  • The completed sequence is: $650, 653, 659, \boxed{671}, \boxed{695}, \boxed{743}$.

• c) 298, 303, 313, \text{___}, \text{___}, \text{___}

  • Pattern: Add 5, then add 10, then add 20, and so on. (The amount added doubles each time).
  • Next numbers: $$313 + 20 = 333$$ $$333 + 40 = 373$$ $$373 + 80 = 453$$
  • The completed sequence is: $298, 303, 313, \boxed{333}, \boxed{373}, \boxed{453}$.

• d) 642, 644, 648, \text{___}, \text{___}, \text{___}

  • Pattern: Add 2, then add 4, then add 8, and so on. (The amount added doubles each time).
  • Next numbers: $$648 + 8 = 656$$ $$656 + 16 = 672$$ $$672 + 32 = 704$$
  • The completed sequence is: $642, 644, 648, \boxed{656}, \boxed{672}, \boxed{704}$.

• e) 589, 593, 601, \text{___}, \text{___}, \text{___}

  • Pattern: Add 4, then add 8, then add 16, and so on. (The amount added doubles each time).
  • Next numbers: $$601 + 16 = 617$$ $$617 + 32 = 649$$ $$649 + 64 = 713$$
  • The completed sequence is: $589, 593, 601, \boxed{617}, \boxed{649}, \boxed{713}$.

• f) 461, 467, 479, \text{___}, \text{___}, \text{___}

  • Pattern: Add 6, then add 12, then add 24, and so on. (The amount added doubles each time).
  • Next numbers: $$479 + 24 = 503$$ $$503 + 48 = 551$$ $$551 + 96 = 647$$
  • The completed sequence is: $461, 467, 479, \boxed{503}, \boxed{551}, \boxed{647}$.

Collecting shells

James collects shells. Every day he picks up twice as many shells as the previous day. On the first day he picks up 7 shells. On the second day he picks up 14. How many shells would he collect on the ninth day?

This is a geometric sequence where the first term $a_1 = 7$ and the common ratio $r = 2$.
The formula for the $n$-th term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$.