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, $S_n$
$1$	$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\, 1 \quad = 1 \quad\;\;\;\;\; = 1^3$
$2$	$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\; 2 + 4 + 2 \quad = 8 \quad\;\;\;\;\; = 2^3$
$3$	$ \qquad\qquad\qquad\qquad\qquad\quad\;\;\;\; 3 + 6 + 9 + 6 + 3 \quad = 27 \quad\;\;\; = 3^3$
$4$	$ \qquad\qquad\qquad\quad\; 4 + 8 + 12 + 16 + 12 + 8 + 4 \quad = 64 \quad\;\;\; = 4^3$
$5$	$ \quad\quad 5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5 \quad = 125 \quad\; = 5^3$
$6$
$7$
$\dots$	$\dots$
$n$	$n + 2n + \dots + (n - 1)n + n^2 + (n - 1)n + \dots + 2n + n \qquad = $ ________

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, \(S_n\)
\(1\)	\( \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\, 1 \quad = 1 \quad\;\;\;\;\; = 1^3\)
\(2\)	\( \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\; 2 + 4 + 2 \quad = 8 \quad\;\;\;\;\; = 2^3\)
\(3\)	\( \qquad\qquad\qquad\qquad\qquad\quad\;\;\;\; 3 + 6 + 9 + 6 + 3 \quad = 27 \quad\;\;\; = 3^3\)
\(4\)	\( \qquad\qquad\qquad\quad\; 4 + 8 + 12 + 16 + 12 + 8 + 4 \quad = 64 \quad\;\;\; = 4^3\)
\(5\)	\( \quad\quad 5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5 \quad = 125 \quad\; = 5^3\)
\(6\)
\(7\)
\(\dots\)	\(\dots\)
\(n\)	\(n + 2n + \dots + (n - 1)n + n^2 + (n - 1)n + \dots + 2n + n \qquad = \) ________

Number Patterns