Consider the following number pattern: State a rule of the numbers at the centre of each row, i.e.,

Question

Consider the following number pattern:

State a rule of the numbers at the centre of each row, i.e., $1, 4, 9, 16, 25, \dots$
Hence find the number at the centre of the $n^{th}$ row.
State a rule for each row of numbers in the pattern above.
Based on your answers in (i) and (ii), complete the following table:

Row Number	Sum of the Numbers of the Row, $S_n$
$1$	$ 1 = 1 = 1^3$
$2$	$ 2 + 4 + 2 = 8 = 2^3$
$3$	$ 3 + 6 + 9 + 6 + 3 = 27 = 3^3$
$4$	$ 4 + 8 + 12 + 16 + 12 + 8 + 4 = 64 = 4^3$
$5$	$ 5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5 = 125 = 5^3$
$6$
$7$
$\dots$	$\dots$
$n$	$2n(1+2+\cdots+(n-1))+n^2=$ ________

Accepted Answer

(i) $n^2$ (ii) Start at $n$, add $n$ to $n^2$, then subtract $n$ back to $n$ (iii) Row 6: $216=6^3$ Row 7: $343=7^3$ $S_n=n^3$

Row Number	Sum of the Numbers of the Row, \(S_n\)
\(1\)	\( 1 = 1 = 1^3\)
\(2\)	\( 2 + 4 + 2 = 8 = 2^3\)
\(3\)	\( 3 + 6 + 9 + 6 + 3 = 27 = 3^3\)
\(4\)	\( 4 + 8 + 12 + 16 + 12 + 8 + 4 = 64 = 4^3\)
\(5\)	\( 5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5 = 125 = 5^3\)
\(6\)
\(7\)
\(\dots\)	\(\dots\)
\(n\)	\(2n(1+2+\cdots+(n-1))+n^2=\) ________

Number Patterns