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Category: Algebra

r-1-n-1-r2-r-

Question Number 3682 by Yozzii last updated on 19/Dec/15 $$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{r}\mathrm{2}^{{r}} }=? \\ $$ Commented by RasheedSindhi last updated on 19/Dec/15 $$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{r}\mathrm{2}^{{r}}…

Question-69201

Question Number 69201 by ahmadshah last updated on 21/Sep/19 Answered by Kunal12588 last updated on 21/Sep/19 $${f}\left({x}\right)=\frac{{x}−\mathrm{5}}{{x}^{\mathrm{2}} −\mathrm{25}}=\frac{{x}−\mathrm{5}}{\left({x}−\mathrm{5}\right)\left({x}+\mathrm{5}\right)} \\ $$$$\Rightarrow\left({x}−\mathrm{5}\right)\left({x}+\mathrm{5}\right)\neq\mathrm{0} \\ $$$$\Rightarrow{x}\neq\mathrm{5},−\mathrm{5} \\ $$$$\therefore{x}\in\mathbb{R}−\left\{−\mathrm{5},\mathrm{5}\right\} \\…

If-a-n-is-sum-of-the-all-primitive-n-th-root-of-unity-Does-n-1-a-n-converge-An-n-th-root-of-unity-say-z-is-primitive-if-it-is-not-k-th-root-of-unity-where-k-lt-n-or-z-n-1-and-z

Question Number 3650 by prakash jain last updated on 18/Dec/15 $$\mathrm{If}\:{a}_{{n}} \:\mathrm{is}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{all}\:\mathrm{primitive}\:{n}^{{th}} \:\mathrm{root}\:\mathrm{of} \\ $$$$\mathrm{unity}. \\ $$$$\mathrm{Does}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{n}} \:\mathrm{converge}? \\ $$$$\mathrm{An}\:{n}^{{th}} \:\mathrm{root}\:\mathrm{of}\:\mathrm{unity},\:\mathrm{say}\:{z},\:\mathrm{is}\:\mathrm{primitive}\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{not} \\ $$$${k}^{{th}}…

If-the-G-M-between-two-numbers-a-b-is-G-and-the-two-A-M-s-between-them-are-p-and-q-then-prove-that-G-2-2p-q-2q-p-

Question Number 3617 by Rasheed Soomro last updated on 16/Dec/15 $${If}\:\:{the}\:{G}.{M}.\:{between}\:{two}\:{numbers}\: \\ $$$${a}\:,\:{b}\:\:\:{is}\:\:{G}\:{and}\:{the}\:{two}\:{A}.{M}.'{s}\:{between} \\ $$$${them}\:{are}\:{p}\:\:{and}\:\:{q}\:,\:{then}\:{prove}\:{that} \\ $$$${G}^{\mathrm{2}} =\left(\mathrm{2}{p}−{q}\right)\left(\mathrm{2}{q}−{p}\right). \\ $$ Commented by Yozzii last updated…

x-4-ax-3-bx-2-cx-d-0-let-x-f-t-linear-perhaps-t-4-At-3-Bt-2-Ct-D-0-can-we-have-4AB-A-3-8C-solving-at-most-a-degree-three-polynomial-

Question Number 69143 by ajfour last updated on 20/Sep/19 $${x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$${let}\:\:{x}={f}\left({t}\right)\:\:{linear}\:{perhaps} \\ $$$${t}^{\mathrm{4}} +{At}^{\mathrm{3}} +{Bt}^{\mathrm{2}} +{Ct}+{D}=\mathrm{0} \\ $$$${can}\:{we}\:{have}\:\: \\ $$$$\:\:\:\mathrm{4}{AB}={A}^{\mathrm{3}} +\mathrm{8}{C}\:\:{solving}\:{at}\:{most}…

1-1-4-6-8-10-12-14-16-18-20-22-S-900-

Question Number 134668 by EDWIN88 last updated on 06/Mar/21 $$\mathrm{1}+\mathrm{1}+\mathrm{4}−\mathrm{6}−\mathrm{8}−\mathrm{10}+\mathrm{12}+\mathrm{14}+\mathrm{16}−\mathrm{18}−\mathrm{20}−\mathrm{22}+… \\ $$$$\mathrm{S}_{\mathrm{900}} \:=\:? \\ $$ Answered by benjo_mathlover last updated on 06/Mar/21 $$\Rightarrow\underset{\mathrm{6}} {\underbrace{\mathrm{1}+\mathrm{1}+\mathrm{4}}}\:\underset{−\mathrm{24}} {\underbrace{−\mathrm{6}−\mathrm{8}−\mathrm{10}}}\:\underset{\mathrm{42}}…