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COMBINING TRIANGLE NUMBERS
This investigation looks at results when adding or subtracting triangle numbers.
Here is a table of the first 21 triangle numbers, $T_1$ to $T_{21}$.
[Table_1]
\(\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} T_1 & T_2 & T_3 & T_4 & T_5 & T_6 & T_7 & T_8 & T_9 & T_{10} & T_{11} & T_{12} & T_{13} & T_{14} & T_{15} & T_{16} & T_{17} & T_{18} & T_{19} & T_{20} & T_{21} \\ \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & 45 & 55 & 66 & 78 & 91 & 105 & 120 & 136 & 153 & 171 & 190 & 210 & 231 \end{array}\)
Find the next two triangle numbers.
\( T_{22} = \text{.......................} \)
\( T_{23} = \text{.............................} \)
[4]
583
580
512
(a) Complete the table.
[Table]
\[
\begin{array}{c|c}
T_1 & 1 \\
\hline
T_2 - T_1 & 2 \\
\hline
T_3 - T_2 & \\
\hline
T_4 - T_3 & \\
\hline
T_5 - T_4 & \\
\hline
T_6 - T_5 & 6 \\
\hline
\cdots & \\
\hline
T_n - T_{n-1} & \\
\end{array}
\]
(b)(i) \( T_n - T_{n-1} = 100 \).
Write down the value of \( n \).
................................................ [1]
(ii) Write down the difference between the 50th and the 49th triangle numbers.
................................................ [1]
524
593
541
Complete the table for adding two consecutive triangle numbers.
[Table_1]
| $T_1$ | 1 |
|------------|---------------|
| $T_2 + T_1$ | 4 |
| $T_3 + T_2$ | 9 |
| $T_4 + T_3$ | |
| $T_5 + T_4$ | |
| $T_6 + T_5$ | |
| $T_n + T_{n-1}$ | |
505
571
553
(a) Use the last row of the table in Question 2(a) to complete the equation $T_n - T_{n-1} = \text{...................}$
Use the last row of the table in Question 3 to complete the equation $T_n + T_{n-1} = \text{...................}$
By adding these two equations together show that $T_n = \frac{n^2+n}{2}$.
(b) Find $T_{1000}$.
571
590
510
(a) The table shows the difference of the squares of two consecutive triangle numbers. Complete the table.
[Table_1]
\[
\begin{array}{|c|c|}
\hline
(T_1)^2 & 1 \\
\hline
(T_2)^2 - (T_1)^2 & 8 \\
\hline
(T_3)^2 - (T_2)^2 & \\
\hline
(T_4)^2 - (T_3)^2 & \\
\hline
(T_5)^2 - (T_4)^2 & 125 \\
\hline
(T_6)^2 - (T_5)^2 & 216 \\
\hline
\cdots & \cdots \\
\hline
(T_n)^2 - (T_{n-1})^2 & \\
\hline
\end{array}
\]
(b) Calculate the difference between the squares of the 50th and the 49th triangle numbers. ....................................................
530
564
549
(a) Start with triangle number $T_5 = 15$ and complete the method of the Example to find another triangle triple.
$T_{15} \; - \; \text{...........} \; = \; \text{...........}$
So $\text{...........} \; - \; \text{...........} \; = \; T_5$
$T_5 \;+ \; \text{...........} \; = \; \text{...........}$
The triangle triple is $(5, \; \text{...........}, \; \text{...........})$
(b) In the table, each row is a triangle triple. Use your answer to part (a) and any patterns you notice to complete the table.
[Table_1]
(c) Use the list of triangle numbers on page 2 to check the triangle triple beginning with 6.
554
551
551
(a) The triangle numbers $T_1$ and $T_3$ are not consecutive. They are two apart. Complete the table for subtracting triangle numbers that are two apart.
\(
\begin{array}{ccc}
T_3 - T_1 & & 5 \\
T_4 - T_2 & & \\
T_5 - T_3 & & \\
T_6 - T_4 & & \\
T_7 - T_5 & & 13 \\
\hdots & & \hdots \\
T_n - T_{n-2} & & \\
\end{array}
\)
(b) Use the triangle number $T_9 = 45$ to find a triangle triple where
- the smallest number is 9
- the difference between the other two numbers is 2.
Use a method similar to that in the Example in Question 6.
(9 , ............. , .............)
590
533
549
