2012 DHS P2 Q4
A finite sequence $\{{{a}_{n}}\}$ has $50$ terms and is such that ${{a}_{n+1}}={{a}_{n}}+0.15$ for $n=1,\,\,2,\,\,3,…..\,\,49$
(i)
Given that ${{a}_{50}}=99{{a}_{1}}$ show that ${{a}_{1}}=0.075$.
[2]
(ii)
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 $m\ge 2$.
(iii)
Determine the smallest value of $k$ such that ${{b}_{k}}<{{a}_{25}}$.
[2]
(iv)
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.
(v)
If $\frac{{{b}_{m}}}{{{b}_{m-1}}}=-0.98$ instead, find $\sum\limits_{m=0}^{\infty }{{{b}_{1+3m}}}$.
[2]
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