1. Pascal's identity states that (+) = (1) + (3) k for all n 0 and...

  

 

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1. Pascal's identity states that (+) = (1) + (3) k for all n 0 and k> 0 where nk. You will find that fact useful in this question. Consider the following equation. (1) = (+) 3 (1) (a) Show that equation (1) holds for n = 2. (b) Suppose P(n) is the assertion that equation (1) holds for a particular value n. By substituting in equation (1), express P(n + 1). (c) Show, by induction on n, that equation (1) is true for all integers n 2, by showing that P(n) P(n+1) for all n > 2. 2. Show, by induction on n, that 9 is a divisor of n+(n+1)+(n+2) for every positive integer n. 1. Pascal's identity states that (+) = (1) + (3) k for all n 0 and k> 0 where nk. You will find that fact useful in this question. Consider the following equation. (1) = (+) 3 (1) (a) Show that equation (1) holds for n = 2. (b) Suppose P(n) is the assertion that equation (1) holds for a particular value n. By substituting in equation (1), express P(n + 1). (c) Show, by induction on n, that equation (1) is true for all integers n 2, by showing that P(n) P(n+1) for all n > 2. 2. Show, by induction on n, that 9 is a divisor of n+(n+1)+(n+2) for every positive integer n.