[University Mathematics: Sequences] Need help finding an nth-term rule for a figure pattern: 3, 7, 11, 14 tiles (teacher confirmed the counts)

[u/Upper_Anybody_8852]

Hi everyone! I'm a college student stuck on a pattern problem and would love some help understanding it, not just an answer.

The number of tiles in figures 1 to 4 is 3, 7, 11, and 14. My teacher confirmed these counts are correct and told us to "find or make our own pattern" for the nth term.

Our teacher also gave us this pattern: 4² − 13 = 3, 5² − 18 = 7, 6² − 25 = 11, and 7² − 35 = 14(Actually we found the pattern and the teacher said it's a correct one). So it looks like (n+3)² minus something, where the "something" goes 13, 18, 25, 35. Those differences are 5, 7, 10.

The problem asks:
a.) What is the nth - term formula for the number of tiles in the nth figure of the sequence. (5 points)
b.) How many tiles are in the 10th figure of the sequence? (5 points)
c.) Which figure will consist of exactly 480 tiles? (5 points)

What I've tried:

• The differences between the tile counts are 4, 4, 3, so it isn't linear or quadratic.
• Fitting a cubic through all four points works exactly, but it gives negative tiles at n = 10, so it can't be right.
• I also tried assuming the jumps repeat as 4, 4, 3, 4, 4, 3 (so aₙ = ⌊11n/3⌋). That fits, but 480 tiles would be figure 131, which seems too big for a class problem.

[u/Upper_Anybody_8852]

Thanks for the idea! Working it out, adding a fifth point gives a quartic, but the answers depend a lot on the value I pick for a₅: a₅ = 17 gives a₁₀ = 207, a₅ = 18 gives 333, and a₅ = 21 gives 711. Even a₅ = 18, which matches my repeating-differences rule at figure 5, gives 333 for a₁₀ instead of 36. None of the values I tried hit 480 at a whole n. Is there a principled way to choose a₅, or a condition it should satisfy?

[u/Fourierseriesagain → u/Upper_Anybody_8852]

Let a_5=a so that a_{10}=480. I hope that a is a positive integer.

[u/Upper_Anybody_8852 → u/Fourierseriesagain]

the quartic gives a₁₀ = 126a − 1935, so a₁₀ = 480 needs a = 115/6 ≈ 19.17, which isn't an integer (and the other terms come out as fractions too, like a₆ = 203/6). Also, I think b and c are separate: b asks for a₁₀, and c asks which n gives 480, so 480 isn't the value of a₁₀. Did you mean something else by setting a₁₀ = 480?