2021 SAJC Revision Package Q2
The sum of the first $n$ terms of a series, ${{S}_{n}}$, is given by $\frac{{{a}^{n}}}{{{2}^{n-1}}}-2$, where $a$ is a non-zero constant and $a\ne 2$.
(i)
Show that ${{T}_{n}}$, the ${{n}^{\text{th}}}$ term of the series, is $\left( a-2 \right){{\left( \frac{a}{2} \right)}^{n-1}}$. Hence show that the given series is a geometric series.
[4]
(ii)
Find the range of values of $a$ for the sum to infinity to exist.
[2]
(iii)
Given that $a=1$, find the least value of $n$ for ${{S}_{n}}$ to be within $\pm 0.2$ of the value of the sum to infinity.
[3]
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