Can we sum from $1000$ to $2000$ excluding multiples of 5?

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Prove that the sum of all the integers between \(m\) and \(n\) inclusive (\(m, n \in \mathbb{Z}_+\), \(n > m\)) is \(\tfrac{1}{2}(m+n)(n-m+1)\). Find the sum of all the integers between \(1000\) and \(2000\) which are not divisible by \(5\).

A geometric series has first term \(2\) and common ratio \(0.95\). The sum of the first \(n\) terms of the series is denoted by \(S_n\) and the sum to infinity is denoted by \(S\). Calculate the least value of \(n\) for which \(S - S_n < 1\).