(i) Expand and simplify $$( a - b ) \left( a ^ { n } + a ^ { n - 1 } b + a ^ { n - 2 } b ^ { 2 } + \cdots + a b ^ { n - 1 } + b ^ { n } \right)$$ (ii) The prime number 3 has the property that it is one less than a square number. Are there any other prime numbers with this property? Justify your answer. (iii) Find all the prime numbers that are one more than a cube number. Justify your answer. (iv) Is $3 ^ { 2015 } - 2 ^ { 2015 }$ a prime number? Explain your reasoning carefully. (v) Is there a positive integer $k$ for which $k ^ { 3 } + 2 k ^ { 2 } + 2 k + 1$ is a cube number? Explain your reasoning carefully. If you require additional space please use the pages at the end of the booklet
(i) [2 marksl] Expanding to $a ^ { n + 1 } + a ^ { n } b + a ^ { n - 1 } b ^ { 2 } + \ldots + a ^ { 2 } b ^ { n - 1 } + a b ^ { n } - a ^ { n } b - a ^ { n - 1 } b ^ { 2 } - \ldots - a b ^ { n } - b ^ { n + 1 }$. Cancelling terms correctly to give $a ^ { n + 1 } - b ^ { n + 1 }$. [0pt]
\section*{2. For ALL APPLICANTS.}
(i) Expand and simplify
$$( a - b ) \left( a ^ { n } + a ^ { n - 1 } b + a ^ { n - 2 } b ^ { 2 } + \cdots + a b ^ { n - 1 } + b ^ { n } \right)$$
(ii) The prime number 3 has the property that it is one less than a square number. Are there any other prime numbers with this property? Justify your answer.\\
(iii) Find all the prime numbers that are one more than a cube number. Justify your answer.\\
(iv) Is $3 ^ { 2015 } - 2 ^ { 2015 }$ a prime number? Explain your reasoning carefully.\\
(v) Is there a positive integer $k$ for which $k ^ { 3 } + 2 k ^ { 2 } + 2 k + 1$ is a cube number? Explain your reasoning carefully.
If you require additional space please use the pages at the end of the booklet