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Question Number 170187 by infinityaction last updated on 18/May/22

Commented by MJS_new last updated on 18/May/22

it′s a number with 609 digits starting with 550355... and ending with ...570632

$$\mathrm{it}'\mathrm{s}\:\mathrm{a}\:\mathrm{number}\:\mathrm{with}\:\mathrm{609}\:\mathrm{digits}\:\mathrm{starting}\:\mathrm{with} \\ $$$$\mathrm{550355}...\:\mathrm{and}\:\mathrm{ending}\:\mathrm{with}\:...\mathrm{570632} \\ $$

Commented by infinityaction last updated on 19/May/22

X+1/x=-1 X²+1/x²=-1 X³+1/x³=2 X²+x+1=0 Multiply both side by x-1 We will get X³-1=0 X³=1 X⁴=x X⁵=x² X⁶=x³=1 So we can say X^(3k+1)=x X^(3k+2)=x² X^3k=1 So X^(3k+1) + 1/x^(3k+1) and x^(3k+2) + 1/x^(3k+2) will get cancel because X^(3k+1) + 1/x^(3k+1) will get even power So -1^(any even term)=1 And X^(3k+2) +1/x^(3k+2) will get odd power because 3k+2= odd term So (-1)^( any odd number)=-1 1+(-1)=0 So now basically we just have to find the sum of (x^3k+1/x^3k)^3k So That will be equal to 2^3k ( for k=1,2,3..... 674) So now will use the formula of sum of gp So sum will be 8(8⁶⁷⁴-1)/7

Commented by infinityaction last updated on 19/May/22

sir please check my solution

$${sir}\:{please}\:{check}\:{my}\:{solution} \\ $$

Answered by Rasheed.Sindhi last updated on 18/May/22

A Try x^2 +x+1=0 (x−1)(x^2 +x+1)=0 x^3 −1=0 x=1,ω, ω^2 (=1/ω) 1 is the root of x−1=0 ∴ x=ω,(1/ω) Σ_(n=1) ^(2022) (x^n +(1/x^n ))^n =? Σ_(n=1) ^3 (x^n +(1/x^n ))^n ={(ω+ω)^1 +(ω^2 +ω^2 )^2 +(ω^3 +ω^3 )^3 } ={2ω+(2ω^2 )^2 +(2ω^3 )^3 } ={2ω+4ω^4 +8} ={2ω+4ω+8} ={6ω+8} =2(3ω+4) .... Σ_(n=4) ^6 (x^n +(1/x^n ))^n ={(ω+ω)^4 +(ω^2 +ω^2 )^5 +(ω^3 +ω^3 )^6 } ={16ω^4 +32ω^(10) +64ω^(18) } ={16ω+32ω+64} =16(3ω+4) Σ_(n=7) ^9 (x^n +(1/x^n ))^n ={(ω+ω)^7 +(ω^2 +ω^2 )^8 +(ω^3 +ω^3 )^9 } ={(2ω)^7 +(2ω^2 )^8 +(2)^9 } =2^7 {(ω)^7 +(2ω^(16) )+(2)^2 } =2^7 {3ω+4} Σ_(n=1) ^(2022) (x^n +(1/x^n ))^n =(3ω+4){2+2^4 +2^7 +...+2^(673) )+(ω^(674) +ω^(674) )^(674) =(3ω+4){2+2^4 +2^7 +...+2^(673) )+(2ω^(674) )^(674) =(3ω+4){2+2^4 +2^7 +...+2^(673) )+2^(674) (ω^(674) )^(674) =(3ω+4){2+2^4 +2^7 +...+2^(673) )+2^(674) ω ..... ...

$$\mathcal{A}\:\mathcal{T}{ry} \\ $$$${x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{3}} −\mathrm{1}=\mathrm{0} \\ $$$${x}=\mathrm{1},\omega,\:\omega^{\mathrm{2}} \left(=\mathrm{1}/\omega\right) \\ $$$$\mathrm{1}\:{is}\:{the}\:{root}\:{of}\:{x}−\mathrm{1}=\mathrm{0} \\ $$$$\therefore\:{x}=\omega,\frac{\mathrm{1}}{\omega} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{2022}} {\sum}}\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{n}} =? \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{n}} \\ $$$$=\left\{\left(\omega+\omega\right)^{\mathrm{1}} +\left(\omega^{\mathrm{2}} +\omega^{\mathrm{2}} \right)^{\mathrm{2}} +\left(\omega^{\mathrm{3}} +\omega^{\mathrm{3}} \right)^{\mathrm{3}} \right\} \\ $$$$=\left\{\mathrm{2}\omega+\left(\mathrm{2}\omega^{\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{2}\omega^{\mathrm{3}} \right)^{\mathrm{3}} \right\} \\ $$$$=\left\{\mathrm{2}\omega+\mathrm{4}\omega^{\mathrm{4}} +\mathrm{8}\right\} \\ $$$$=\left\{\mathrm{2}\omega+\mathrm{4}\omega+\mathrm{8}\right\} \\ $$$$=\left\{\mathrm{6}\omega+\mathrm{8}\right\} \\ $$$$=\mathrm{2}\left(\mathrm{3}\omega+\mathrm{4}\right) \\ $$$$.... \\ $$$$\underset{{n}=\mathrm{4}} {\overset{\mathrm{6}} {\sum}}\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{n}} \\ $$$$=\left\{\left(\omega+\omega\right)^{\mathrm{4}} +\left(\omega^{\mathrm{2}} +\omega^{\mathrm{2}} \right)^{\mathrm{5}} +\left(\omega^{\mathrm{3}} +\omega^{\mathrm{3}} \right)^{\mathrm{6}} \right\} \\ $$$$=\left\{\mathrm{16}\omega^{\mathrm{4}} +\mathrm{32}\omega^{\mathrm{10}} +\mathrm{64}\omega^{\mathrm{18}} \right\} \\ $$$$=\left\{\mathrm{16}\omega+\mathrm{32}\omega+\mathrm{64}\right\} \\ $$$$=\mathrm{16}\left(\mathrm{3}\omega+\mathrm{4}\right) \\ $$$$\underset{{n}=\mathrm{7}} {\overset{\mathrm{9}} {\sum}}\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{n}} \\ $$$$=\left\{\left(\omega+\omega\right)^{\mathrm{7}} +\left(\omega^{\mathrm{2}} +\omega^{\mathrm{2}} \right)^{\mathrm{8}} +\left(\omega^{\mathrm{3}} +\omega^{\mathrm{3}} \right)^{\mathrm{9}} \right\} \\ $$$$=\left\{\left(\mathrm{2}\omega\right)^{\mathrm{7}} +\left(\mathrm{2}\omega^{\mathrm{2}} \right)^{\mathrm{8}} +\left(\mathrm{2}\right)^{\mathrm{9}} \right\} \\ $$$$=\mathrm{2}^{\mathrm{7}} \left\{\left(\omega\right)^{\mathrm{7}} +\left(\mathrm{2}\omega^{\mathrm{16}} \right)+\left(\mathrm{2}\right)^{\mathrm{2}} \right\} \\ $$$$=\mathrm{2}^{\mathrm{7}} \left\{\mathrm{3}\omega+\mathrm{4}\right\} \\ $$$$ \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{2022}} {\sum}}\left({x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{n}} \\ $$$$=\left(\mathrm{3}\omega+\mathrm{4}\right)\left\{\mathrm{2}+\mathrm{2}^{\mathrm{4}} +\mathrm{2}^{\mathrm{7}} +...+\mathrm{2}^{\mathrm{673}} \right)+\left(\omega^{\mathrm{674}} +\omega^{\mathrm{674}} \right)^{\mathrm{674}} \\ $$$$=\left(\mathrm{3}\omega+\mathrm{4}\right)\left\{\mathrm{2}+\mathrm{2}^{\mathrm{4}} +\mathrm{2}^{\mathrm{7}} +...+\mathrm{2}^{\mathrm{673}} \right)+\left(\mathrm{2}\omega^{\mathrm{674}} \right)^{\mathrm{674}} \\ $$$$=\left(\mathrm{3}\omega+\mathrm{4}\right)\left\{\mathrm{2}+\mathrm{2}^{\mathrm{4}} +\mathrm{2}^{\mathrm{7}} +...+\mathrm{2}^{\mathrm{673}} \right)+\mathrm{2}^{\mathrm{674}} \left(\omega^{\mathrm{674}} \right)^{\mathrm{674}} \\ $$$$=\left(\mathrm{3}\omega+\mathrm{4}\right)\left\{\mathrm{2}+\mathrm{2}^{\mathrm{4}} +\mathrm{2}^{\mathrm{7}} +...+\mathrm{2}^{\mathrm{673}} \right)+\mathrm{2}^{\mathrm{674}} \omega \\ $$$$..... \\ $$$$... \\ $$

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