Triangular Numbers Practice :: Teacher Resources and Classroom ... - Free Printable

Let's solve the Triangular Numbers Practice problem step by step.

---

🔷 What are Triangular Numbers?

Triangular numbers are numbers that can form an equilateral triangle when represented with dots. The pattern starts as:

- 1 (one dot)
- 3 (three dots forming a triangle)
- 6 (six dots forming a larger triangle)
- 10, 15, etc.

Each number is the sum of the natural numbers up to a certain point.

The formula for the n-th triangular number is:

$$
T_n = \frac{n(n+1)}{2}
$$

---

We are given a table with two columns:
- Left: $ n $ and $ T_n $
- Right: $ T_n $ and we need to find $ n $

So, we have to:
1. Use the formula to fill in missing values.
2. For known $ T_n $, solve for $ n $ using the formula.

---

✔ Step 1: Fill in the missing $ T_n $ values from the left side

| $ n $ | $ T_n = \frac{n(n+1)}{2} $ |
|--------|-------------------------------|
| 1 | $ \frac{1(2)}{2} = 1 $ |
| 2 | $ \frac{2(3)}{2} = 3 $ |
| 3 | $ \frac{3(4)}{2} = 6 $ |
| 4 | $ \frac{4(5)}{2} = 10 $ |
| 5 | $ \frac{5(6)}{2} = 15 $ |
| 6 | $ \frac{6(7)}{2} = 21 $ |
| 7 | $ \frac{7(8)}{2} = 28 $ |
| 8 | $ \frac{8(9)}{2} = 36 $ |
| 9 | $ \frac{9(10)}{2} = 45 $ |
| 10 | $ \frac{10(11)}{2} = 55 $ |

✔ So the left column completes as:

| $ n $ | $ T_n $ |
|--------|----------|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
| 5 | 15 |
| 6 | 21 |
| 7 | 28 |
| 8 | 36 |
| 9 | 45 |
| 10 | 55 |

---

✔ Step 2: Solve for $ n $ given $ T_n $ on the right side

We're given $ T_n $ values and must find $ n $. We use the formula:

$$
T_n = \frac{n(n+1)}{2}
\Rightarrow n(n+1) = 2T_n
$$

We'll solve this quadratic equation for each value.

---

#### 1. $ T_n = 20 $

$$
n(n+1) = 40 \\
n^2 + n - 40 = 0
$$

Use quadratic formula:

$$
n = \frac{-1 \pm \sqrt{1 + 160}}{2} = \frac{-1 \pm \sqrt{161}}{2} \approx \frac{-1 \pm 12.7}{2}
$$

Positive root: $ \approx 5.85 $ → Not integer → ✘ No solution? Wait — but 20 isn't a triangular number.

Wait! Let's check if 20 is actually a triangular number.

Try $ n = 5 $: $ T_5 = 15 $
$ n = 6 $: $ T_6 = 21 $ → So 20 is not a triangular number

But the table says "fill in the empty boxes" — maybe it's asking us to find which $ n $ gives that $ T_n $, or perhaps only valid ones?

But let’s look at the right-hand column:

| $ T_n $ | $ n $ |
|---------|-------|
| 20 | ? |
| 25 | ? |
| 40 | ? |
| 50 | ? |
| 100 | ? |
| 200 | ? |
| 500 | ? |
| 1000 | ? |
| 5000 | ? |
| 25000 | ? |

We need to find $ n $ such that $ T_n = \frac{n(n+1)}{2} $ matches these values.

But some may not be triangular numbers.

Let’s test each one:

---

#### 🔹 $ T_n = 20 $

Solve $ \frac{n(n+1)}{2} = 20 $

→ $ n(n+1) = 40 $

Try $ n=6 $: $ 6×7 = 42 $ → too big
$ n=5 $: $ 5×6 = 30 $ → too small
No integer solution → Not a triangular number

So no valid $ n $

But the task says “discover the correct answers” — so maybe it's okay to leave blank?

But likely, the values are meant to be triangular numbers. Let's check if they’re close.

Wait — maybe there’s a typo? Or perhaps we should find the closest $ n $?

But more likely, the intention is to solve $ \frac{n(n+1)}{2} = T_n $ and see which ones work.

Let’s go through each.

---

🔍 Solve for $ n $ given $ T_n $

We’ll use approximation:

$$
n(n+1)/2 \approx n^2/2 \Rightarrow n \approx \sqrt{2T_n}
$$

Then try nearby integers.

---

#### 1. $ T_n = 20 $

$ \sqrt{2×20} = \sqrt{40} \approx 6.3 $

Try $ n=6 $: $ T_6 = 21 $ → too big
$ n=5 $: $ T_5 = 15 $ → too small
→ Not triangular

✘ No solution

---

#### 2. $ T_n = 25 $

$ \sqrt{50} \approx 7.1 $

Try $ n=7 $: $ T_7 = 28 $
$ n=6 $: $ T_6 = 21 $
No match → ✘ Not triangular

---

#### 3. $ T_n = 40 $

$ \sqrt{80} \approx 8.9 $

Try $ n=8 $: $ T_8 = 36 $
$ n=9 $: $ T_9 = 45 $ → too big
No match → ✘ Not triangular

---

#### 4. $ T_n = 50 $

$ \sqrt{100} = 10 $

Try $ n=10 $: $ T_{10} = 55 $
$ n=9 $: $ T_9 = 45 $
No → ✘ Not triangular

---

#### 5. $ T_n = 100 $

$ \sqrt{200} \approx 14.1 $

Try $ n=14 $: $ T_{14} = \frac{14×15}{2} = 105 $
$ n=13 $: $ \frac{13×14}{2} = 91 $
No → ✘ Not triangular

---

#### 6. $ T_n = 200 $

$ \sqrt{400} = 20 $

Try $ n=20 $: $ T_{20} = \frac{20×21}{2} = 210 $
$ n=19 $: $ \frac{19×20}{2} = 190 $
No → ✘ Not triangular

---

#### 7. $ T_n = 500 $

$ \sqrt{1000} \approx 31.6 $

Try $ n=31 $: $ \frac{31×32}{2} = 496 $
$ n=32 $: $ \frac{32×33}{2} = 528 $
Closest: 496 → not 500 → ✘ Not triangular

---

#### 8. $ T_n = 1000 $

$ \sqrt{2000} \approx 44.7 $

Try $ n=44 $: $ \frac{44×45}{2} = 990 $
$ n=45 $: $ \frac{45×46}{2} = 1035 $
No → ✘ Not triangular

---

#### 9. $ T_n = 5000 $

$ \sqrt{10000} = 100 $

Try $ n=100 $: $ \frac{100×101}{2} = 5050 $
$ n=99 $: $ \frac{99×100}{2} = 4950 $
No → ✘ Not triangular

---

#### 10. $ T_n = 25000 $

$ \sqrt{50000} \approx 223.6 $

Try $ n=223 $: $ \frac{223×224}{2} = 223×112 = 24976 $
$ n=224 $: $ \frac{224×225}{2} = 112×225 = 25200 $

24976 vs 25000 → very close, but not equal → ✘ Not triangular

---

🚩 Conclusion: None of the listed $ T_n $ values (20, 25, 40, ..., 25000) are actual triangular numbers!

That suggests either:
- There's a mistake in the worksheet
- Or the task is to find the closest $ n $ such that $ T_n $ is near the given number
- Or we're supposed to find $ n $ such that $ T_n = \text{value} $, even if it doesn’t exist

But since the instructions say: *"use the formula to discover the correct answers"*, and give $ T_n $, we assume the values are meant to be triangular.

Wait — let’s double-check the left side: it goes up to $ T_{10} = 55 $

But the right side has large values like 25000.

Perhaps the right column is asking for $ n $ such that $ T_n = $ the given number, and we must solve for $ n $ using the formula.

But as shown, none of those are triangular numbers.

Wait — could it be that the right column is $ T_n $, and we need to find $ n $? But then we can't.

Alternatively, maybe the table is incomplete, and we need to fill in both sides?

Looking back at the image:

There are two tables:
- Left: $ n $ and $ T_n $ — filled up to $ n=10 $
- Right: $ T_n $ and $ n $ — with $ T_n $ given, and $ n $ blank

But none of the $ T_n $ values are triangular.

Unless... maybe the values are meant to be solved via inverse formula?

Let’s suppose we want to find $ n $ such that $ T_n = x $, i.e., solve:

$$
\frac{n(n+1)}{2} = x \Rightarrow n^2 + n - 2x = 0
$$

Then:

$$
n = \frac{-1 + \sqrt{1 + 8x}}{2}
$$

Let’s compute this for each $ x $:

---

✔ Compute $ n $ for each $ T_n $ using the formula

$$
n = \frac{-1 + \sqrt{1 + 8T_n}}{2}
$$

We'll compute and see if it's close to an integer.

---

#### 1. $ T_n = 20 $

$ \sqrt{1 + 160} = \sqrt{161} \approx 12.688 $

$ n = (-1 + 12.688)/2 = 11.688/2 = 5.844 $ → not integer

---

#### 2. $ T_n = 25 $

$ \sqrt{1 + 200} = \sqrt{201} \approx 14.177 $

$ n = (13.177)/2 = 6.588 $ → no

---

#### 3. $ T_n = 40 $

$ \sqrt{321} \approx 17.916 $ → $ n = (16.916)/2 = 8.458 $

---

#### 4. $ T_n = 50 $

$ \sqrt{401} \approx 20.025 $ → $ n = 19.025/2 = 9.5125 $

---

#### 5. $ T_n = 100 $

$ \sqrt{801} \approx 28.3 $ → $ n = 27.3/2 = 13.65 $

---

#### 6. $ T_n = 200 $

$ \sqrt{1601} \approx 40.012 $ → $ n = 39.012/2 = 19.506 $

---

#### 7. $ T_n = 500 $

$ \sqrt{4001} \approx 63.25 $ → $ n = 62.25/2 = 31.125 $

---

#### 8. $ T_n = 1000 $

$ \sqrt{8001} \approx 89.44 $ → $ n = 88.44/2 = 44.22 $

---

#### 9. $ T_n = 5000 $

$ \sqrt{40001} \approx 200.0025 $ → $ n = 199.0025/2 = 99.501 $

Very close to 100 — but not exact

---

#### 10. $ T_n = 25000 $

$ \sqrt{200001} \approx 447.21 $ → $ n = 446.21/2 = 223.105 $

---

Now, notice that:

- $ T_{100} = 5050 $
- $ T_{223} = 24976 $
- $ T_{224} = 25200 $

So none of the listed values are triangular numbers.

But wait — maybe the values are meant to be approximations, or perhaps it's a typo?

Alternatively, perhaps the right column is asking for $ n $ such that $ T_n $ is close to the given number?

Or maybe the intention is to practice the formula, even if values aren’t perfect?

But the instruction says: "fill in the empty boxes using the formula to discover the correct answers"

So likely, the values are meant to be triangular numbers.

Wait — what if the right column is $ T_n $, and we are to find $ n $ such that $ T_n $ is that number?

But again, none are.

Unless...

Wait — let’s reverse-engineer.

Maybe the given $ T_n $ are correct, and we just need to solve for $ n $ using the formula, and accept decimal answers?

But the context is elementary math — probably expecting integer answers.

Alternatively, maybe the values are wrong, and we should calculate $ T_n $ for $ n=20,25,... $?

But that doesn't make sense.

Wait — another idea: perhaps the right column is $ n $, and the left is $ T_n $? But no — the table shows:

| $ T_n $ | $ n $ |
|---------|-------|
| 20 | ? |

So $ T_n $ is given, $ n $ is to be found.

But none are triangular.

Wait — unless the values are meant to be $ n $, and we calculate $ T_n $?

But the header says: "Here is a table of triangular numbers. Please fill in the empty boxes..."

And the first column is $ T_n $, second is $ n $.

So likely, it's asking: for each $ T_n $, find $ n $.

But since none are triangular, maybe the task is to find the $ n $ that makes $ T_n $ closest?

But that’s speculative.

---

✔ Alternative Interpretation

Perhaps the right column is not $ T_n $, but rather $ n $, and we need to compute $ T_n $?

But the table shows:

| $ T_n $ | $ n $ |
|---------|-------|
| 20 | ? |

So it’s clear: $ T_n $ is given, $ n $ is unknown.

But since none are triangular, maybe it's a trick question?

Or perhaps the values are meant to be solved approximately?

But let’s think differently.

Wait — what if the right column is $ n $, and the left is $ T_n $? That would make more sense.

But the label says: "a table of triangular numbers", and the first column is labeled $ T_n $, second is $ n $.

But in standard tables, $ n $ is on the left.

Wait — looking at the image:

> Here is a table of triangular numbers. Please fill in the empty boxes using the formula to discover the correct answers.

Then it shows:

| $ T_n $ | $ n $ |
|---------|-------|
| 20 | ? |

But this implies $ T_n = 20 $, find $ n $

But 20 is not triangular.

Unless the first column is $ n $, and second is $ T_n $, but the headers are swapped?

Wait — let’s look carefully.

In the left table, it’s clearly:

| $ n $ | $ T_n $ |
|--------|----------|
| 1 | 1 |
| 2 | 3 |
| ... | ... |

So $ n $ is first, $ T_n $ is second.

Then the right table has:

| $ T_n $ | $ n $ |
|---------|-------|
| 20 | ? |

So it's reversed.

But that doesn't make sense.

Unless the right table is also $ n $ and $ T_n $, but the labels are misaligned?

Wait — perhaps the right table is meant to be $ n $ and $ T_n $, but the column headers are off.

Alternatively, maybe the right column is $ T_n $, and we are to find $ n $ such that $ T_n = $ that value.

But again, none are triangular.

---

🧠 Final Possibility: The values are meant to be $ n $, and we calculate $ T_n $

Suppose the first column is $ n $, and the second is $ T_n $, but the header says "$ T_n $" and "$ n $", which is backwards.

But that would make sense.

For example:

If $ n = 20 $, then $ T_n = \frac{20×21}{2} = 210 $

Similarly:
- $ n = 25 $: $ T_n = \frac{25×26}{2} = 325 $
- $ n = 40 $: $ T_n = \frac{40×41}{2} = 820 $
- $ n = 50 $: $ T_n = 1275 $
- $ n = 100 $: $ T_n = 5050 $
- $ n = 200 $: $ T_n = 20100 $
- $ n = 500 $: $ T_n = 125250 $
- $ n = 1000 $: $ T_n = 500500 $
- $ n = 5000 $: $ T_n = 12,502,500 $
- $ n = 25000 $: $ T_n = 312,575,000 $

But the table shows $ T_n $ in the first column, so this doesn't fit.

---

✔ Best Explanation

There is a mistake in the worksheet: the values in the right column (20, 25, 40, ...) are not triangular numbers, so there is no integer $ n $ such that $ T_n = $ those values.

Therefore, the task cannot be completed as written.

However, if the intent is to practice the formula, we can:

1. For each given $ T_n $, use the formula $ n = \frac{-1 + \sqrt{1 + 8T_n}}{2} $ to find approximate $ n $
2. Or, realize that the values are not triangular, and leave blanks

But since the worksheet says "discover the correct answers", likely the values are meant to be triangular.

Let’s check if any of them are close to triangular numbers:

- $ T_{10} = 55 $
- $ T_{20} = 210 $
- $ T_{50} = 1275 $
- $ T_{100} = 5050 $
- $ T_{200} = 20100 $
- $ T_{500} = 125250 $

None of the given values (20, 25, 40, 50, 100, 200, 500, 1000, 5000, 25000) are triangular.

But 25000 is close to $ T_{223} = 24976 $, $ T_{224} = 25200 $

So perhaps the intended answer is to find $ n $ such that $ T_n $ is closest to the given value.

But that’s not what the formula does.

---

✔ Final Answer

After careful analysis, none of the given $ T_n $ values (20, 25, 40, 50, 100, 200, 500, 1000, 5000, 25000) are triangular numbers, so there is no integer $ n $ that satisfies $ T_n = $ those values.

However, if the task is to use the formula to solve for $ n $, we can do so symbolically:

For $ T_n = x $, solve:
$$
n = \frac{-1 + \sqrt{1 + 8x}}{2}
$$

Then plug in each value:

| $ T_n $ | $ n $ (approximate) |
|---------|---------------------|
| 20 | 5.84 |
| 25 | 6.59 |
| 40 | 8.46 |
| 50 | 9.51 |
| 100 | 13.65 |
| 200 | 19.51 |
| 500 | 31.12 |
| 1000 | 44.22 |
| 5000 | 99.50 |
| 25000 | 223.10 |

These are the closest $ n $ values, but not exact.

---

🟩 Recommendation

The worksheet likely contains a typo. The values in the right column should be actual triangular numbers, such as:

- 21 ($ T_6 $)
- 28 ($ T_7 $)
- 36 ($ T_8 $)
- 45 ($ T_9 $)
- 55 ($ T_{10} $)
- 66 ($ T_{11} $)
- 78 ($ T_{12} $)
- etc.

Or, alternatively, the first column should be $ n $, and second $ T_n $.

But as it stands, the task cannot be completed with integer answers.

---

✔ Summary

- The formula for triangular numbers is $ T_n = \frac{n(n+1)}{2} $
- The given $ T_n $ values (20, 25, 40, ...) are not triangular
- Therefore, no integer $ n $ exists for any of them
- The worksheet may have errors in the values provided
- If allowed, you can compute approximate $ n $ using $ n = \frac{-1 + \sqrt{1 + 8T_n}}{2} $

---

If you have access to the original source, I recommend checking for typos. Otherwise, inform the teacher about the inconsistency.

Let me know if you'd like a corrected version of the table!