Math worksheet on triangular numbers, part of a series on special sequences including triangle numbers, square numbers, and Fibonacci.

Let’s go step by step through each question.

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1) Can you draw the next two terms in the space above?

We are shown:
- 1st triangular number: 1 dot (just one)
- 2nd: 3 dots (triangle with 2 rows: 1 + 2)
- 3rd: 6 dots (triangle with 3 rows: 1 + 2 + 3)

So, to get the next ones:

→ 4th triangular number = 1 + 2 + 3 + 4 = 10 dots
Draw a triangle with 4 rows: top row 1, then 2, then 3, then 4 at the bottom.

→ 5th triangular number = 1 + 2 + 3 + 4 + 5 = 15 dots
Draw a triangle with 5 rows: 1, 2, 3, 4, 5 from top to bottom.

*(You’d draw these in the blank space above — but since we’re text-based, just know how many dots and how they’re arranged.)*

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2) What are the first 5 triangular numbers?

From above:

1st: 1
2nd: 1+2 = 3
3rd: 1+2+3 = 6
4th: 1+2+3+4 = 10
5th: 1+2+3+4+5 = 15

✔ First 5: 1, 3, 6, 10, 15

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3) In words, describe how the sequence increases from term to term.

Each time, we add the next counting number.

From 1st to 2nd: add 2 → 1 + 2 = 3
From 2nd to 3rd: add 3 → 3 + 3 = 6
From 3rd to 4th: add 4 → 6 + 4 = 10
From 4th to 5th: add 5 → 10 + 5 = 15

So: To get the next triangular number, add the position number of that term.
Or simpler: Add 2, then 3, then 4, then 5, etc., increasing by 1 each time.

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4) Here is part of the sequence for triangular numbers: 78, 91, 105, ... What number comes next?

Let’s check the differences:

91 - 78 = 13
105 - 91 = 14

So the pattern of adding is increasing by 1 each time: +13, then +14, so next should be +15

105 + 15 = 120

✔ Next number: 120

*(Check: Is 120 a triangular number? Let’s see: n(n+1)/2 = 120 → n² + n - 240 = 0 → solve: n = [-1 ± √(1+960)]/2 = [-1±√961]/2 = [-1±31]/2 → positive root: 30/2=15 → yes! 15th triangular number is 120. Correct.)*

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5) Vinny says that the sequences of triangular numbers is an arithmetic sequence. Do you agree with Vinny? Explain your answer.

An arithmetic sequence has a constant difference between terms.

Triangular numbers: 1, 3, 6, 10, 15, 21,...

Differences:
3 - 1 = 2
6 - 3 = 3
10 - 6 = 4
15 - 10 = 5
21 - 15 = 6

The differences are increasing: 2, 3, 4, 5, 6... not constant.

So ✘ No, it is not an arithmetic sequence.

✔ Answer: No, because the difference between terms keeps changing — it increases by 1 each time. Arithmetic sequences must have the same difference every time.

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6) The nth term of the sequence is ½n(n + 1):

a) Find the 56th triangular number.

Plug n = 56 into formula:

T₅₆ = ½ × 56 × (56 + 1) = ½ × 56 × 57

First, 56 × 57:

56 × 50 = 2800
56 × 7 = 392
Total = 2800 + 392 = 3192

Now half of that: 3192 ÷ 2 = 1596

✔ 56th triangular number = 1596

b) Is 842 a triangular number?

Set ½n(n+1) = 842

Multiply both sides by 2:
n(n+1) = 1684

So: n² + n - 1684 = 0

Use quadratic formula:
n = [-1 ± √(1 + 4×1684)] / 2
= [-1 ± √(1 + 6736)] / 2
= [-1 ± √6737] / 2

What’s √6737?

Try 82² = 6724
83² = 6889 → too big

82² = 6724 → 6737 - 6724 = 13 → not a perfect square.

So √6737 is not whole → n is not whole → 842 is not a triangular number.

✔ Answer: No, 842 is not a triangular number.

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7) If you find the sum of 2 consecutive triangular numbers you get another type of number. What are these numbers? Show how you get your answers.

Let’s try small examples:

1st and 2nd: T₁ = 1, T₂ = 3 → 1 + 3 = 4 → which is 2² → square!

2nd and 3rd: T₂ = 3, T₃ = 6 → 3 + 6 = 9 → 3² → square!

3rd and 4th: T₃ = 6, T₄ = 10 → 6 + 10 = 16 → 4² → square!

4th and 5th: T₄ = 10, T₅ = 15 → 10 + 15 = 25 → 5² → square!

Pattern: Sum of Tₙ and Tₙ₊₁ = (n+1)²

Why?

Formula:
Tₙ = ½n(n+1)
Tₙ₊₁ = ½(n+1)(n+2)

Sum = ½n(n+1) + ½(n+1)(n+2)
Factor out ½(n+1):
= ½(n+1)[n + (n+2)]
= ½(n+1)(2n + 2)
= ½(n+1) × 2(n + 1)
= (n+1)(n+1)
= (n+1)²

✔ So the sum of two consecutive triangular numbers is always a square number.

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Final Answer:

1) Draw triangles with 10 dots (4 rows) and 15 dots (5 rows).
2) First 5 triangular numbers: 1, 3, 6, 10, 15
3) Each term adds the next integer: +2, +3, +4, +5, etc.
4) Next number after 78, 91, 105 is 120
5) No — differences aren’t constant; they increase by 1 each time.
6a) 56th triangular number: 1596
6b) No, 842 is not a triangular number.
7) Sum of two consecutive triangular numbers gives a square number — proven algebraically and with examples.