Question
Statement 1 If the terms of a GP are alternately positive and negative, then the common ratio of such a GP has to be a negative number andStatement 2 nth term of the GP a $+a r+a r^{2}+\ldots$ is ar $^{n-1}$
Step 1
The nth term of a geometric progression (GP) is given by $a r^{n-1}$, where $a$ is the first term and $r$ is the common ratio. This is a well-known formula and is correct. Show more…
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Statement 1 $\mathrm{S}_{r}$ is the sum of an infinite $\mathrm{G} . \mathrm{P}$ with 1st term $\mathrm{r}$ and common ratio $\frac{1}{\mathrm{r}+1}$. Then $\mathrm{S}_{\mathrm{r}}-\mathrm{r}$ depends on $\mathrm{r}$. and Statement 2 Sum of an infinite geometric series the common ratio is numerically less than 1 exists only if $\div \mathrm{r} \div<1$
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For a geometric sequence with first term $a_{1}$ and common ratio $r,$ where $r \neq 0, r \neq 1,$ the sum of the first $n$ terms is $S_{n}=a_{1} \cdot \frac{1-r^{n}}{1-r}$
Sequences; Induction; the Binomial Theorem
Geometric Sequences; Geometric Series
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