$a, 1, b$ are three consecutive terms of an arithmetic series, and $b, a$ and $\frac{8}{3}$ are the first three terms of an infinite geometric series that has a sum of $S$. find $a, b$ and $S$.
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[Math] $a, 1, b$ are three consecutive terms of an arithmetic series, find $a, b$ and $S$ of the infinite geometric series
arithmetic-progressionsgeometric seriessequences-and-series
Best Answer
by $a, 1, b$ are three consecutive terms of an arithmetic series we can know that:$$a+b=2$$ by $b, a$ and $\frac{8}{3}$ are the first three terms of an infinite geometric series we can know that:$$\frac{8}{3}b=a^2$$ so we got such an equation set:$$\begin{cases}a+b=2\\\frac{8}{3}b=a^2\\\end{cases}$$ the solution of above equation set is:$\begin{cases}a_1=\frac43\\b_1=\frac23\end{cases}$ or $\begin{cases}a_2=-4\\b_2=6\end{cases}$
The first case
the geometric ratio of this geometric sequence $q_1=2$ and $S$ is infinite in this case
The second case
the geometric ratio of this geometric sequence $q_2=-\frac23$ and $S=\lim_{n\to +\infty}\frac{b_2(1-q_2^n)}{1-q_2}=\lim_{n\to +\infty}\frac{6(1-({-\frac23})^n)}{1-(-\frac23)}=\frac{18}{5}$