Sequence and Series - Problems involving inequalities

A series of greenhouse frames is made using metal bars and connectors.
Figure \(n\) consists of \(n\) rows and \(n+1\) columns of identical rectangular sections. Adjacent sections share a metal bar. A connector is placed at every point where the bars meet.
The first three figures can be represented as follows, where each box represents one rectangular section.

Figure 1

Figure 2

Figure 3

1. Draw Figure \(4\).

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2. Complete the table.

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   Figure	Number of Bars	Number of Connectors
   \(1\)	\(7\)	\(6\)
   \(2\)	\(17\)	\(12\)
   \(3\)	\(31\)	\(20\)
   \(4\)
   \(5\)

3. Is it possible for a figure in this series to contain exactly \(100\) bars? Explain your answer.

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4. 1. Write down, in terms of \(n\), an expression for the number of connectors in Figure \(n\).

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   2. A builder has \(420\) connectors and an unlimited number of bars. Find the largest figure that can be constructed.

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