Question and Answers Forum

All Questions Topic List

AllQuestion and Answers: Page 208

Question Number 198187 Answers: 1 Comments: 1

Question Number 198184 Answers: 1 Comments: 0

Question Number 198182 Answers: 1 Comments: 0

Question Number 198186 Answers: 1 Comments: 0

Question Number 198178 Answers: 2 Comments: 0

f(xf(y)+x)=xy+f(x) f:R→R f(x)=?

$${f}\left({xf}\left({y}\right)+{x}\right)={xy}+{f}\left({x}\right) \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 198176 Answers: 1 Comments: 0

Question Number 198175 Answers: 1 Comments: 0

Prove The following Functional equation: ζ(x,s)=((2Γ(1−s))/((2π)^((1−s)) )){sin(((πs)/2))Σ_(m=1) ^∞ [((cos(2πmx))/m^((1−s)) )]+cos(((πs)/2))Σ_(m=1) ^∞ [((sin(2πmx))/m^((1−s)) )]}

$${Prove}\:{The}\:{following}\:{Functional}\:{equation}: \\ $$$$\zeta\left({x},{s}\right)=\frac{\mathrm{2}\Gamma\left(\mathrm{1}−{s}\right)}{\left(\mathrm{2}\pi\right)^{\left(\mathrm{1}−{s}\right)} }\left\{{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{cos}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]+{cos}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{sin}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]\right\} \\ $$

Question Number 198166 Answers: 3 Comments: 0

if f(x)=x^2 +bx+c f(f(1))=f(f(2))=0 and f(1)≠f(2) find f(0)=?

$${if}\:{f}\left({x}\right)={x}^{\mathrm{2}} +{bx}+{c} \\ $$$${f}\left({f}\left(\mathrm{1}\right)\right)={f}\left({f}\left(\mathrm{2}\right)\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)\neq{f}\left(\mathrm{2}\right) \\ $$$${find}\:{f}\left(\mathrm{0}\right)=? \\ $$

Question Number 198161 Answers: 1 Comments: 0

Question Number 198158 Answers: 1 Comments: 0

Question Number 198152 Answers: 2 Comments: 0

a_(n+2) = (√(a_n ×a_(n+1) )) ∀ n≥1 , n ∈ N and here a_(1 ) = α and a_2 = β then prove that lim_(n→∞) a_(n+2) = (α×β^2 )^(1/3)

$$\:\:\:\:{a}_{{n}+\mathrm{2}} \:=\:\:\:\sqrt{{a}_{{n}} ×{a}_{{n}+\mathrm{1}} }\:\:\:\forall\:{n}\geqslant\mathrm{1}\:,\:{n}\:\in\:\mathrm{N} \\ $$$$\:\mathrm{and}\:\mathrm{here}\:\:\mathrm{a}_{\mathrm{1}\:} =\:\alpha\:\:{and}\:{a}_{\mathrm{2}} =\:\beta\:\:\mathrm{then} \\ $$$$\:\:\:\mathrm{prove}\:\mathrm{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}+\mathrm{2}} \:\:\:=\:\:\left(\alpha×\beta^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} \\ $$

Question Number 198151 Answers: 1 Comments: 0

Question Number 198147 Answers: 1 Comments: 0

if a,x,y,b is an AP and a,p,q,b is a GP. prove that xy≥pq. (with a, b >0)

$${if}\:{a},{x},{y},{b}\:{is}\:{an}\:{AP}\:{and}\:{a},{p},{q},{b}\:{is}\:{a}\:{GP}. \\ $$$${prove}\:{that}\:{xy}\geqslant{pq}. \\ $$$$\left({with}\:{a},\:{b}\:>\mathrm{0}\right) \\ $$

Question Number 198146 Answers: 0 Comments: 1

Please suggest youtube playlist to prepare one for mathematics olympiad. Thanks in advance.

$${Please}\:{suggest}\:{youtube}\:{playlist}\:{to} \\ $$$${prepare}\:{one}\:{for}\:{mathematics}\:{olympiad}. \\ $$$${Thanks}\:{in}\:{advance}. \\ $$$$ \\ $$

Question Number 198156 Answers: 1 Comments: 0

Prove that ((2t−1)/(lnt−ln(1−t)))=∫^( 1) _( 0) t^x (1−t)^(1−x) dx and ∫^( 1) _( 0) ((2t−1)/(lnt−ln(1−t)))dt = (π/2)∫^( 1) _( 0) ((x(1−x))/(sin(πx)))dx

$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}=\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{t}^{\mathrm{x}} \left(\mathrm{1}−\mathrm{t}\right)^{\mathrm{1}−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{and}\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}\mathrm{dt}\:\:=\:\:\frac{\pi}{\mathrm{2}}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{sin}\left(\pi\mathrm{x}\right)}\mathrm{dx} \\ $$

Question Number 198141 Answers: 1 Comments: 0

∫^( 1) _( 0) ((x(1−x))/(sin(πx)))dx=???

$$\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \:\frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{sin}\left(\pi\mathrm{x}\right)}\mathrm{dx}=??? \\ $$

Question Number 198136 Answers: 1 Comments: 0