Linear Patterns: Discovering the Constant Step
Watch a tile pattern grow, count the new tiles, and discover the formula 2n − 1 hiding in the picture
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). An L-shaped staircase made of glowing amber-coloured square tiles rises and extends into the distance, getting wider and taller stage by stage. Each new stage of the staircase adds exactly two more tiles than the previous — one on top, one on the side. The staircase emerges from a dark misty foreground and ascends into a deep starry navy sky. The image conveys: every new stage adds the same fixed amount — the visual signature of a linear pattern. Painterly cinematic illustration. Dark background. No text labels.
Stage 1 of a tile pattern has 1 tile. Stage 2 has 3 tiles. Stage 3 has 5 tiles. Stage 4 has 7 tiles. How many tiles in stage 100? In stage 1,000?
Each stage adds exactly 2 more tiles than the previous. Can you find a formula in that gives the count for stage directly?
From Mahāvīra's Gaṇita-sāra-saṅgraha — On Figurate Numbers
एकद्व्यादिक्रमेण आसन् वर्धमानानां यथा
ता एव सङ्ख्या त्रिकोणाः चतुष्कोणाः च गण्यन्ते॥
(eka-dvy-ādi-krameṇa āsan vardhamānānāṃ yathā / tā eva saṃkhyā tri-koṇāḥ catuṣ-koṇāḥ ca gaṇyante)
'जो संख्याएँ एक-दो-तीन के क्रम में बढ़ती हैं — उन्हीं को त्रिकोणीय (triangular) और चतुष्कोणीय (square) संख्याएँ कहते हैं।'
'Numbers that grow in the order one, two, three — these very numbers are called triangular and square (figurate) numbers.'
Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850 CE) had a whole chapter on figurate numbers — numbers that arise from arrangements of dots, tiles, or pebbles in geometric shapes. Triangular numbers (), square numbers (), pentagonal numbers — Indian mathematicians counted them all, 800 years before Pascal's triangle reached Europe. The L-staircase you'll meet on this page is in the same family: a growing pattern whose count follows a simple algebraic rule.
The growing L-staircase
Look at the four stages of the tile pattern:
- Stage 1: 1 square.
- Stage 2: 3 squares (an L-shape).
- Stage 3: 5 squares.
- Stage 4: 7 squares.
The sequence of counts is — the odd numbers.
How is each stage built from the previous one? Look carefully: each stage takes the previous shape and adds exactly 2 squares — one on top of the vertical column, one on the right of the horizontal row. The pattern grows in a steady L-shape rhythm, and the count goes up by 2 every stage.
AI Generation Prompt
Educational mathematics illustration showing four stages of a growing tile pattern, side by side, evenly spaced. Stage 1: a single pale-orange square at the baseline. Stage 2: three pale-orange squares forming an L-tromino — the base square plus one square to its right and one square stacked on top. Stage 3: five pale-orange squares forming a larger L. Stage 4: seven pale-orange squares forming an even larger L. Each square has a thin darker orange border. Below each L is a centred caption: 'Stage 1', 'Stage 2', 'Stage 3', 'Stage 4'. Style: clean educational textbook illustration. Dark background, orange accent labels, clean technical illustration style.
Loading simulator…
From "next is +2" to "the nth term is "
Knowing each stage is 2 more than the previous lets us continue the sequence: . But this is not enough by itself — to find stage 100, we'd still have to count 100 jumps of 2.
We want a formula that gives the count directly in terms of the stage number . Let's hunt for it:
| Stage | Count | Pattern |
|---|---|---|
| 1 | 1 | ✓ |
| 2 | 3 | ✓ |
| 3 | 5 | ✓ |
| 4 | 7 | ✓ |
| 5 | 9 | ✓ |
The rule is count = . This is a linear polynomial in : degree 1, leading coefficient 2, constant term .
The constant difference of 2 between consecutive counts and the leading coefficient 2 in the formula are the same number — that's no coincidence. The leading coefficient of a linear polynomial is exactly the per-step change.
Looking at the picture of the L-staircase, what tells you the formula must have a +1 step per stage (linear), rather than (say) a ×2 step per stage (geometric, like 1, 2, 4, 8, …)?
Practice Yourself — Find the nth Term
Each sequence is a linear pattern. Find the nth-term formula (an expression in ) and use it to compute the 20th term. Cover the answers and try.
- (constant pattern)
Answers: 1. , . 2. , . 3. , . 4. , . 5. , . 6. (constant — degree 0, technically not linear but the constant-difference test still works with difference 0).
Recipe: the leading coefficient is the constant difference. The constant term is found by checking : if and we want this to match the first term, then .
Q1.If a linear pattern has nth term , what is the 20th term?
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). An L-shaped staircase made of glowing amber-coloured square tiles rises and extends into the distance, getting wider and taller stage by stage. Each new stage of the staircase adds exactly two more tiles than the previous — one on top, one on the side. The staircase emerges from a dark misty foreground and ascends into a deep starry navy sky. The image conveys: every new stage adds the same fixed amount — the visual signature of a linear pattern. Painterly cinematic illustration. Dark background. No text labels.
Stage 1 of a tile pattern has 1 tile. Stage 2 has 3 tiles. Stage 3 has 5 tiles. Stage 4 has 7 tiles. How many tiles in stage 100? In stage 1,000?
Each stage adds exactly 2 more tiles than the previous. Can you find a formula in that gives the count for stage directly?
From Mahāvīra's Gaṇita-sāra-saṅgraha — On Figurate Numbers
एकद्व्यादिक्रमेण आसन् वर्धमानानां यथा
ता एव सङ्ख्या त्रिकोणाः चतुष्कोणाः च गण्यन्ते॥
(eka-dvy-ādi-krameṇa āsan vardhamānānāṃ yathā / tā eva saṃkhyā tri-koṇāḥ catuṣ-koṇāḥ ca gaṇyante)
'जो संख्याएँ एक-दो-तीन के क्रम में बढ़ती हैं — उन्हीं को त्रिकोणीय (triangular) और चतुष्कोणीय (square) संख्याएँ कहते हैं।'
'Numbers that grow in the order one, two, three — these very numbers are called triangular and square (figurate) numbers.'
Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850 CE) had a whole chapter on figurate numbers — numbers that arise from arrangements of dots, tiles, or pebbles in geometric shapes. Triangular numbers (), square numbers (), pentagonal numbers — Indian mathematicians counted them all, 800 years before Pascal's triangle reached Europe. The L-staircase you'll meet on this page is in the same family: a growing pattern whose count follows a simple algebraic rule.
The growing L-staircase
Look at the four stages of the tile pattern:
- Stage 1: 1 square.
- Stage 2: 3 squares (an L-shape).
- Stage 3: 5 squares.
- Stage 4: 7 squares.
The sequence of counts is — the odd numbers.
How is each stage built from the previous one? Look carefully: each stage takes the previous shape and adds exactly 2 squares — one on top of the vertical column, one on the right of the horizontal row. The pattern grows in a steady L-shape rhythm, and the count goes up by 2 every stage.
AI Generation Prompt
Educational mathematics illustration showing four stages of a growing tile pattern, side by side, evenly spaced. Stage 1: a single pale-orange square at the baseline. Stage 2: three pale-orange squares forming an L-tromino — the base square plus one square to its right and one square stacked on top. Stage 3: five pale-orange squares forming a larger L. Stage 4: seven pale-orange squares forming an even larger L. Each square has a thin darker orange border. Below each L is a centred caption: 'Stage 1', 'Stage 2', 'Stage 3', 'Stage 4'. Style: clean educational textbook illustration. Dark background, orange accent labels, clean technical illustration style.
Loading simulator…
From "next is +2" to "the nth term is "
Knowing each stage is 2 more than the previous lets us continue the sequence: . But this is not enough by itself — to find stage 100, we'd still have to count 100 jumps of 2.
We want a formula that gives the count directly in terms of the stage number . Let's hunt for it:
| Stage | Count | Pattern |
|---|---|---|
| 1 | 1 | ✓ |
| 2 | 3 | ✓ |
| 3 | 5 | ✓ |
| 4 | 7 | ✓ |
| 5 | 9 | ✓ |
The rule is count = . This is a linear polynomial in : degree 1, leading coefficient 2, constant term .
The constant difference of 2 between consecutive counts and the leading coefficient 2 in the formula are the same number — that's no coincidence. The leading coefficient of a linear polynomial is exactly the per-step change.
Looking at the picture of the L-staircase, what tells you the formula must have a +1 step per stage (linear), rather than (say) a ×2 step per stage (geometric, like 1, 2, 4, 8, …)?
Q1.If a linear pattern has nth term , what is the 20th term?