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{\bf Question}
\begin{description}
\item[(a)] Write down a formula for the sum of then odd numbers between
1 and $2n - 1$ inclusive.
\item[(b)] The first term in a convergent geometric sequence is 1
and the total sum of the series is 2. What is the second term of
the sequence?
\end{description}
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{\bf Answer}
\begin{description}
\item[(a)]
Arithmetic series:
$u_1 = 1, \ \ \ u_2 = 3, \ \ \ u_m = 1 + 2(m-1), \ \ \ u_n = 1 +
2(n - 1) = 2n - 1$ Hence there are $n$ terms in the series.
Use the formula for summing arithmetic series:
$S_n = \frac{1}{2}n(u_1 + u_n) = \frac{1}{2}n(2n - 1 + 1) =
\frac{1}{2}n \times 2n = n^2$
\item[(b)]
The sum of a geometric series with constant $r$ and first term 1
is:
$\ds S = \sum_{n=1}^\infty u_n = 1 + r + r^2 + r^3 + ... =
\frac{1}{(1 -r)}$
$\ds S = \frac{1}{(1 - r)} = 2$
Hence $\ds 1 - r = \frac{1}{2} \Rightarrow r = \frac{1}{2}
\Rightarrow u_2 = r = \frac{1}{2}$
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