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An irrigation system is composed of pieces of pipe, where each piece has $8$8​ holes through which water is expelled. When the pieces are laid out and joined together, each piece added covers two holes on the previous piece, preventing water from passing through.a. Write a rule for the number of holes that water can pass through when $n$n​ pieces of pipe are joined together.$a_n=$an​=​

Question

An irrigation system is composed of pieces of pipe, where each piece has 888​ holes through which water is expelled. When the pieces are laid out and joined together, each piece added covers two holes on the previous piece, preventing water from passing through.a. Write a rule for the number of holes that water can pass through when nnn​ pieces of pipe are joined together.an=a_n=an​=​

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Solution

The rule for the number of holes that water can pass through when nn pieces of pipe are joined together can be written as:

an=8n2(n1)a_n = 8n - 2(n-1)

Here's the step by step explanation:

  1. Each piece of pipe has 8 holes, so if you have nn pieces, you initially have 8n8n holes.

  2. However, when you join the pipes together, each new piece covers 2 holes from the previous piece. So, for nn pieces, you have 2(n1)2(n-1) holes covered. We subtract 1 from nn because the first piece of pipe doesn't cover any holes.

  3. Therefore, the total number of holes that water can pass through is the initial number of holes minus the number of covered holes, which gives us 8n2(n1)8n - 2(n-1).

This problem has been solved

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