Exam-Style Question on Sequences

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Question id: 131. This question is similar to one that appeared on a GCSE Higher paper (specimen) for 2017. The use of a calculator is allowed.

The diagrams below show a sequence of patterns made from red and yellow tiles.

(a) Find an expression, in terms of $$n$$, for the number of red tiles in pattern $$n$$.

The total number of red and yellow tiles in each pattern is always the sum of the squares of two consecutive whole numbers.

(b) Find an expression, in terms of $$n$$, for the total number tiles in Pattern $$n$$ giving your answer in its simplest form.

(c) Is there a pattern for which the total number of tiles is 303? Give a reason for your answer.

(d) Explain why the total number of tiles in any pattern of this sequence is always an odd number.

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