Show algebraically that any square number is the sum of
two consecutive triangular numbers.

We want to show algebraically that any square number can be expressed as the sum of two consecutive triangular numbers.

### Triangular Numbers:
The formula for the \(n\)-th triangular number, \(T_n\), is:

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

### Statement to Prove:
We need to show that for any square number, \(n^2\), it can be written as the sum of two consecutive triangular numbers. Specifically, we want to prove:

\[
n^2 = T_n + T_{n-1}
\]

where \(T_n\) is the \(n\)-th triangular number and \(T_{n-1}\) is the \((n-1)\)-th triangular number.

### Step-by-Step Proof:

1. **Write the formula for \(T_n\) and \(T_{n-1}\):**
   Using the formula for the triangular numbers:
   \[
   T_n = \frac{n(n+1)}{2}
   \]
   \[
   T_{n-1} = \frac{(n-1)n}{2}
   \]

2. **Add the two triangular numbers:**
   We now add \(T_n\) and \(T_{n-1}\):
   \[
   T_n + T_{n-1} = \frac{n(n+1)}{2} + \frac{(n-1)n}{2}
   \]

3. **Simplify the expression:**
   Combine the terms under a common denominator:
   \[
   T_n + T_{n-1} = \frac{n(n+1) + (n-1)n}{2}
   \]
   Distribute \(n\) inside both terms:
   \[
   T_n + T_{n-1} = \frac{n^2 + n + n^2 - n}{2}
   \]
   Simplify further:
   \[
   T_n + T_{n-1} = \frac{2n^2}{2}
   \]

4. **Final result:**
   The right-hand side simplifies to:
   \[
   T_n + T_{n-1} = n^2
   \]

Thus, we have shown that any square number \(n^2\) is indeed the sum of two consecutive triangular numbers.

### Conclusion:
\[
n^2 = T_n + T_{n-1}
\]

This completes the algebraic proof.

---

Do you want further details or have any questions?

Here are 5 related questions that extend this concept:
1. How does the formula for triangular numbers relate to other polygonal numbers?
2. Can you prove this result geometrically?
3. What happens if you sum non-consecutive triangular numbers?
4. How can you use triangular numbers to find the sum of the first \(n\) natural numbers?
5. How can this concept be generalized to other number patterns?

**Tip:** Whenever you are dealing with sums of numbers following a particular pattern, using their explicit formulas is a good way to start your proof.

Show algebraically that any square number is the sum of two consecutive triangular numbers.

Explore how to prove that any square number can be expressed as the sum of two consecutive triangular numbers with an algebraic step-by-step proof. This problem covers key algebraic concepts related to triangular and square numbers. Suitable for students in Grades 9-12.

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