Sum of First n Natural Numbers via Staircase Model
What you will learn
Derive and verify the arithmetic progression formula for the sum of the first n natural numbers (S_n = n(n+1)/2) using a physical 2D staircase model of unit square tiles.
Materials needed
- Cardboard sheet (30 cm cm) with 1 cm cm grid × 1What is it?Cardboard sheet (30 cm cm) with 1 cm cm grid × 1₹50
Wooden interlocking arithmetic progression staircase model with 110 magnetic unit cubes.
Specifications- Base Size
- 200 mm × 220 mm magnetic display board
- Cubes Included
- 55 red magnetic cubes + 55 yellow magnetic cubes (15 mm edge)
- Storage
- Fitted wooden tray with sliding lid
- Unit wooden / plastic square tiles (1 cm cm)What is it?Unit wooden / plastic square tiles (1 cm cm)₹95
Square grid millimeter graph sheets (1 cm × 1 cm printed unit squares).
Specifications- Grid Size
- 10 mm × 10 mm bold grid squares
- Paper
- 120 GSM colored craft paper (red and yellow)
- Colored chart paper (red and yellow) eachWhat is it?Colored chart paper (red and yellow) each₹140
Safety guillotine paper trimmer and non-toxic adhesive stick.
Specifications- Trimmer
- Steel blade safety paper trimmer with alignment grid
- Glue
- PVA acid-free non-staining stick
- Scissors, Glue stick, and marker pen setWhat is it?Scissors, Glue stick, and marker pen set₹185
Standard laboratory apparatus used in Sum of First n Natural Numbers via Staircase Model.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
Tick items as you collect them. Tap + to see the product, specifications and buy options.
Stay safe
- All unit tiles must have identical square dimensions (1 cm × 1 cm) for gapless interlocking.
- Align tiles strictly along rectangular grid lines.
- Ensure student clearly distinguishes between the two contrasting colored staircases.
- Test with multiple values of n (n = 4, n = 5, n = 10) to confirm generalizability.
Steps
- 1Choose n = 6 as the target natural number.Observation
Cut 55 unit squares (1 cm × 1 cm) from red squared paper and 55 identical unit squares from yellow squared paper.
SafetyUse guillotine paper trimmer with safety guard for cutting identical unit squares.
- 2Construct a staircase of red unit squares: 1 square in column 1, 2 in column 2, 3 in column 3, up to 6 in column 6. Total red tiles = 1 + 2 + 3 + 4 + 5 + 6 = 21.Observation
Arrange red squares into 10 columns of heights 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 to build a stepped staircase shape.
SafetyKeep rows and columns aligned with square grid lines.
- 3Construct an identical second staircase using yellow unit squares (total 21 yellow tiles).Observation
Assemble the 55 yellow squares into an identical inverted staircase shape.
SafetyVerify that both staircases contain exactly 55 squares each.
- 4Invert the yellow staircase and interlock it with the red staircase.Observation
Interlock the yellow staircase upside-down with the red staircase: they mesh seamlessly into a single rectangle.
SafetyObserve that the stepped profiles fit together with zero gaps.
- 5Observe that the two interlocked staircases form a perfect solid rectangle.Observation
Measure dimensions of resulting rectangle: height has 10 squares, base has (10 + 1) = 11 squares.
SafetyTotal squares in combined rectangle = 10 × 11 = 110 unit squares.
- 6Count the dimensions of the rectangle: breadth = n = 6 rows, length = n + 1 = 7 columns.Observation
Since the rectangle comprises two equal staircases, each staircase contains 110 / 2 = 55 squares, verifying Sn = n(n+1)/2.
SafetyPaste completed composite rectangle onto laboratory folder chart.
- 7Total area of rectangle = n × (n+1) = 6 × 7 = 42 tiles. Since it comprises two identical staircases, the sum of one staircase is S_n = n(n+1)/2 = 42/2 = 21.
Tap + on a step to see what you will observe and any safety tips.
Record your observations
| Number of Terms (n) | Staircase Sum Expression | Theoretical Formula n(n+1)/2 | Physical Unit Squares Counted | Rectangle Dimensions (n × (n+1)) |
|---|---|---|---|---|
| n = 1 | 1 | 1 × 2 / 2 = 1 | 1 unit square | 1 × 2 = 2 (Half = 1) |
| n = 3 | 1 + 2 + 3 | 3 × 4 / 2 = 6 | 6 unit squares | 3 × 4 = 12 (Half = 6) |
| n = 5 | 1 + 2 + 3 + 4 + 5 | 5 × 6 / 2 = 15 | 15 unit squares | 5 × 6 = 30 (Half = 15) |
| n = 8 | 1 + 2 + ... + 8 | 8 × 9 / 2 = 36 | 36 unit squares | 8 × 9 = 72 (Half = 36) |
| n = 10 | 1 + 2 + 3 + ... + 10 | 10 × 11 / 2 = 55 | 55 unit squares | 10 × 11 = 110 (Half = 55) |
Tap a highlighted cell to write your own observation.
Result and conclusion
Two identical staircases of n = 6 steps (1+2+3+4+5+6 = 21 tiles each) fit together to form a rectangle of dimensions n × (n+1) = 6 × 7 = 42 tiles. Hence S_n = 42 / 2 = 21.
| Observation | What it means |
|---|---|
| Geometric Staircase Representation | Summing consecutive integers 1 to n corresponds to stacking columns of unit squares of heights 1, 2, ..., n into a stepped staircase. |
| Interlocking Rectangular Pairing | Two identical n-step staircases (colored blue and yellow) interlock perfectly to form an n × (n + 1) rectangle. |
| Algebraic Formula Derivation | Total area of rectangle = n × (n + 1); therefore, area of single staircase = 1/2 · n(n + 1), proving Gauss' formula geometrically. |
| Arithmetic Progression Foundation | Provides the visual and geometric bedrock for standard arithmetic progression AP sum Sn = n/2 [2a + (n-1)d] with a=1, d=1. |
Sample readings. Replace with your own results.