Polygonal Numbers: Build Sequences and Derive nth-Term Formulas

Algebra and Function Practice

Polygonal numbers count dots arranged in regular polygons that grow outward layer by layer. Starting from a single dot, each new layer adds a fixed number of dots: 2 for triangles, 3 for squares, 4 for pentagons, and 5 for hexagons. This constant layer increment makes the sequences quadratic, so each has a closed-form formula of the form an^2 + bn + c. In this lesson you will build the first several terms from dot diagrams, tabulate them, use first and second differences to identify the quadratic structure, and derive the nth-term formula for each polygon family. You will also verify formulas against counted terms and learn to avoid common errors such as misindexing the first term or confusing the layer increment with the common difference of the sequence itself.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Building polygonal-number sequences from dot diagrams

A polygonal number is the total count of dots in a regular polygonal arrangement with a given number of layers. The first term, for n = 1, is always a single dot, so every polygonal sequence begins with 1. To move from term n to term n+1, add a new outer layer. For an s-sided polygon, each new layer adds s - 2 dots more than the previous layer added, starting from an increment of s - 2 for the first step. Equivalently, the increment from term n to term n+1 is (s - 2)n + 1.

For triangular numbers (s = 3), the increments are 2, 3, 4, 5, ... giving terms 1, 3, 6, 10, 15, ... For square numbers (s = 4), increments are 3, 5, 7, 9, ... giving 1, 4, 9, 16, 25, ... For pentagonal numbers (s = 5), increments are 4, 7, 10, 13, ... giving 1, 5, 12, 22, 35, ... For hexagonal numbers (s = 6), increments are 5, 9, 13, 17, ... giving 1, 6, 15, 28, 45, ...

A dot diagram for term n shows nested polygonal rings. For a square, term 3 is a 3-by-3 array of dots (9 dots). For a pentagon, term 3 has a central dot, a ring of 4 dots, and an outer ring of 7 dots, totaling 12. Always count the dots in each ring and sum them to confirm the sequence term.

1. Draw term 1 as a single dot.
2. For term 2, add the first ring: s - 2 dots for an s-sided polygon.
3. For each subsequent term, add a ring with 3 more dots than the previous ring for triangles, or generally s - 2 more dots than the previous ring.
4. Count all dots in the diagram and record the total in a table.
5. Repeat until you have at least five terms.

Using differences to identify the quadratic form

List the terms in a table with n = 1, 2, 3, 4, 5. Compute the first differences (term n+1 minus term n). For polygonal numbers, the first differences increase by a constant amount, s - 2, each time. Compute the second differences (difference of first differences). The second difference is constant and equals s - 2.

A constant second difference means the nth term is a quadratic expression an^2 + bn + c. The constant second difference equals 2a, so a = (s - 2)/2. For triangles, s - 2 = 1, so a = 1/2. For squares, a = 1. For pentagons, a = 3/2. For hexagons, a = 2.

Substitute n = 1 and n = 2 into an^2 + bn + c and set equal to the known first two terms to solve for b and c. Because every polygonal sequence starts with 1, c is determined once a and b are known.

1. Tabulate terms for n = 1 to 5.
2. Compute first differences.
3. Compute second differences and confirm they are constant.
4. Set 2a equal to the constant second difference and solve for a.
5. Use the first two terms to form two equations and solve for b and c.
6. Write the closed-form formula and test it on a term you counted.

Deriving and verifying the closed-form formulas

For triangular numbers, a = 1/2. Using T1 = 1 and T2 = 3: (1/2)(1)^2 + b(1) + c = 1 and (1/2)(2)^2 + b(2) + c = 3. Subtracting gives b = 1 and c = 0, so Tn = n(n+1)/2. Check n = 4: 4 times 5 divided by 2 equals 10, matching the counted term.

For square numbers, a = 1. Using S1 = 1 and S2 = 4: 1 + b + c = 1 and 4 + 2b + c = 4. Subtracting gives b = 0 and c = 0, so Sn = n^2. Check n = 5: 25, matching the counted term.

For pentagonal numbers, a = 3/2. Using P1 = 1 and P2 = 5: 3/2 + b + c = 1 and 6 + 2b + c = 5. Subtracting gives b = 1/2 and c = 0, so Pn = (3n^2 - n)/2. Check n = 4: (48 - 4)/2 = 22, matching the counted term.

For hexagonal numbers, a = 2. Using H1 = 1 and H2 = 6: 2 + b + c = 1 and 8 + 2b + c = 6. Subtracting gives b = -1 and c = 0, so Hn = 2n^2 - n. Check n = 5: 50 - 5 = 45, matching the counted term.

In every case, verify by substituting a term index you counted from the dot diagram. If the formula does not match, recheck the second difference and the first two terms.

1. Write the general quadratic an^2 + bn + c.
2. Substitute a from the second difference.
3. Use n = 1 and n = 2 to solve for b and c.
4. State the formula clearly with the variable n defined as the term number.
5. Verify with at least one counted term beyond n = 2.

Worked example

Build the first five pentagonal numbers from dot diagrams, compute first and second differences, and derive the nth-term formula. Verify the formula for n = 5.

Four practice exercises

Common mistakes

• Starting the sequence at n = 0 instead of n = 1. Polygonal numbers conventionally begin with term 1 equal to 1 dot. Keep n = 1 as the first term unless a problem explicitly states otherwise.
• Confusing the layer increment with the first difference of the sequence. The layer increment for an s-sided polygon is s - 2 dots more than the previous layer. The first difference between consecutive terms is (s - 2)n + 1, which changes with n. The second difference is the constant s - 2.
• Assuming the constant second difference equals a instead of 2a. For a quadratic an^2 + bn + c, the second difference is 2a. Divide the constant second difference by 2 to find a.
• Miscounting dots in a diagram by missing the central dot or double-counting shared vertices. Count ring by ring from the center outward and record each ring count before summing. For a pentagon, term 3 rings are 1, 4, and 7, totaling 12.

Self-check rubric

• Dot diagrams correctly show nested rings with counts matching the stated terms.
• Table lists at least five terms with n = 1 through 5.
• First differences are computed correctly and increase by a constant amount.
• Second difference is identified as constant and equal to s - 2.
• Value of a is correctly set to half the second difference.
• Equations from n = 1 and n = 2 are solved correctly for b and c.
• Closed-form formula is stated with n defined as the term number.
• Formula is verified against at least one counted term beyond n = 2.
• Handwriting and diagram conventions are consistent and legible; alternative valid layouts are accepted if counts are correct.

Questions and answers