2012 DHS P2 Q4

A finite sequence \(\{{{a}_{n}}\}\) has \(50\) terms and is such that \({{a}_{n+1}}={{a}_{n}}+0.15\) for \(n=1,\,\,2,\,\,3,.....\,\,49\)

1. Given that \({{a}_{50}}=99{{a}_{1}}\) show that \({{a}_{1}}=0.075\).[2]
2. Find, without using a calculator, the value of \(\sum\limits_{n=1}^{50}{{{a}_{n}}}\).[2]

Another infinite sequence \(\{{{b}_{m}}\}\) is such that \({{b}_{1}}={{a}_{50}}\) and \(\frac{{{b}_{m}}}{{{b}_{m-1}}}=0.98\) for \(m\ge 2\).

3. Determine the smallest value of \(k\) such that \({{b}_{k}}<{{a}_{25}}\).[2]
4. Find the least value of \(h\) such that the sum of the first \(h\) terms of \(\{{{b}_{m}}\}\) is more than \(99%\) of its sum to infinity.[2]
5. If \(\frac{{{b}_{m}}}{{{b}_{m-1}}}=-0.98\) instead, find \(\sum\limits_{m=0}^{\infty }{{{b}_{1+3m}}}\).[2]