If first differences are not constant, compare the terms with familiar patterns such as n2, or count a rectangular arrangement. Justify the formula from the structure wherever possible. Example A rectangle of dots has n rows and n+1 dots in each row. The totals 2,6,12,20, follow T_n=n(n+1). At position 8, there are 8 x 9=72 dots. The sequence 3,6,11,18, is 2 above 1,4,9,16,, so its stated square-number pattern has rule T_n=n2+2. Check each given position before using the rule.

If first differences are not constant, compare the terms with familiar patterns such as \(n^2\), or count a rectangular arrangement. Justify the formula from the structure wherever possible.

A rectangle of dots has \(n\) rows and \(n+1\) dots in each row. The totals \(2,6,12,20,\ldots\) follow \(T_n=n(n+1)\). At position \(8\), there are \(8\times9=72\) dots.

The sequence \(3,6,11,18,\ldots\) is \(2\) above \(1,4,9,16,\ldots\), so its stated square-number pattern has rule \(T_n=n^2+2\). Check each given position before using the rule.

Patterns involving squares and products

Patterns involving squares and products