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Make-r-subject-formula-S-n-a-1-r-n-1-r-




Question Number 139104 by I want to learn more last updated on 22/Apr/21
Make  r  subject formula:     S_n   =   ((a(1   −   r^n ))/(1  −   r))
$$\mathrm{Make}\:\:\mathrm{r}\:\:\mathrm{subject}\:\mathrm{formula}:\:\:\:\:\:\mathrm{S}_{\mathrm{n}} \:\:=\:\:\:\frac{\mathrm{a}\left(\mathrm{1}\:\:\:−\:\:\:\mathrm{r}^{\mathrm{n}} \right)}{\mathrm{1}\:\:−\:\:\:\mathrm{r}} \\ $$
Commented by mr W last updated on 22/Apr/21
you can′t! there is no formula for  the roots of n^(th)  degree polynomial  equation.
$${you}\:{can}'{t}!\:{there}\:{is}\:{no}\:{formula}\:{for} \\ $$$${the}\:{roots}\:{of}\:{n}^{{th}} \:{degree}\:{polynomial} \\ $$$${equation}. \\ $$
Commented by I want to learn more last updated on 22/Apr/21
Thanks sir, i appreciate
$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{i}\:\mathrm{appreciate} \\ $$
Commented by I want to learn more last updated on 23/Apr/21
Can we find  r  here?         r^n    =   rθ   +  β
$$\mathrm{Can}\:\mathrm{we}\:\mathrm{find}\:\:\mathrm{r}\:\:\mathrm{here}?\:\:\:\:\:\:\:\:\:\mathrm{r}^{\mathrm{n}} \:\:\:=\:\:\:\mathrm{r}\theta\:\:\:+\:\:\beta \\ $$
Commented by MJS_new last updated on 23/Apr/21
no.  r^n +pr+q=0  is solveable for n∈Z∧−4≤n≤4 but not generally
$$\mathrm{no}. \\ $$$$\mathrm{r}^{{n}} +{pr}+{q}=\mathrm{0} \\ $$$$\mathrm{is}\:\mathrm{solveable}\:\mathrm{for}\:{n}\in\mathbb{Z}\wedge−\mathrm{4}\leqslant{n}\leqslant\mathrm{4}\:\mathrm{but}\:\mathrm{not}\:\mathrm{generally} \\ $$
Answered by MJS_new last updated on 22/Apr/21
(S_n /a)=(((1−r)(1+r+r^2 +...+r^(n−1) ))/(1−r))  r^(n−1) +r^(n−2) +...r^1 +1−(S_n /a)=0  further steps impossible
$$\frac{{S}_{{n}} }{{a}}=\frac{\left(\mathrm{1}−{r}\right)\left(\mathrm{1}+{r}+{r}^{\mathrm{2}} +…+{r}^{{n}−\mathrm{1}} \right)}{\mathrm{1}−{r}} \\ $$$${r}^{{n}−\mathrm{1}} +{r}^{{n}−\mathrm{2}} +…{r}^{\mathrm{1}} +\mathrm{1}−\frac{{S}_{{n}} }{{a}}=\mathrm{0} \\ $$$$\mathrm{further}\:\mathrm{steps}\:\mathrm{impossible} \\ $$
Commented by I want to learn more last updated on 23/Apr/21
Thanks sir.
$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$

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