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a-n-number-series-a-1-5-d-3-a-2-2-a-1-2-a-4-2-a-3-2-a-6-2-a-5-2-a-10-2-a-9-2-




Question Number 207262 by hardmath last updated on 10/May/24
a_n  - number series  a_1  = 5  d = 3  a_2 ^2 −a_1 ^2  + a_4 ^2 −a_3 ^2  + a_6 ^2 −a_5 ^2  + ... + a_(10) ^2 −a_9 ^2  = ?
$$\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:-\:\mathrm{number}\:\mathrm{series} \\ $$$$\mathrm{a}_{\mathrm{1}} \:=\:\mathrm{5} \\ $$$$\mathrm{d}\:=\:\mathrm{3} \\ $$$$\mathrm{a}_{\mathrm{2}} ^{\mathrm{2}} −\mathrm{a}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{a}_{\mathrm{4}} ^{\mathrm{2}} −\mathrm{a}_{\mathrm{3}} ^{\mathrm{2}} \:+\:\mathrm{a}_{\mathrm{6}} ^{\mathrm{2}} −\mathrm{a}_{\mathrm{5}} ^{\mathrm{2}} \:+\:…\:+\:\mathrm{a}_{\mathrm{10}} ^{\mathrm{2}} −\mathrm{a}_{\mathrm{9}} ^{\mathrm{2}} \:=\:? \\ $$
Answered by A5T last updated on 10/May/24
(a_2 −a_1 )(a_2 +a_1 )=d×[2a_1 +d]  a_(2n) ^2 −a_(2n−1) ^2 =(a_(2n) −a_(2n−1) )(a_(2n) +a_(2n−1) )  =d×[a_1 +(2n−1)d+a_1 +(2n−2)d]  =d×[2a_1 +d(4n−3)]  Σ_(n=1) ^5 a_(2n) ^2 −a_(2n−1) ^2 =d[10a_1 +d(1+5+9+13+17)]  =3[50+3(45)]=555
$$\left({a}_{\mathrm{2}} −{a}_{\mathrm{1}} \right)\left({a}_{\mathrm{2}} +{a}_{\mathrm{1}} \right)={d}×\left[\mathrm{2}{a}_{\mathrm{1}} +{d}\right] \\ $$$${a}_{\mathrm{2}{n}} ^{\mathrm{2}} −{a}_{\mathrm{2}{n}−\mathrm{1}} ^{\mathrm{2}} =\left({a}_{\mathrm{2}{n}} −{a}_{\mathrm{2}{n}−\mathrm{1}} \right)\left({a}_{\mathrm{2}{n}} +{a}_{\mathrm{2}{n}−\mathrm{1}} \right) \\ $$$$={d}×\left[{a}_{\mathrm{1}} +\left(\mathrm{2}{n}−\mathrm{1}\right){d}+{a}_{\mathrm{1}} +\left(\mathrm{2}{n}−\mathrm{2}\right){d}\right] \\ $$$$={d}×\left[\mathrm{2}{a}_{\mathrm{1}} +{d}\left(\mathrm{4}{n}−\mathrm{3}\right)\right] \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\sum}}{a}_{\mathrm{2}{n}} ^{\mathrm{2}} −{a}_{\mathrm{2}{n}−\mathrm{1}} ^{\mathrm{2}} ={d}\left[\mathrm{10}{a}_{\mathrm{1}} +{d}\left(\mathrm{1}+\mathrm{5}+\mathrm{9}+\mathrm{13}+\mathrm{17}\right)\right] \\ $$$$=\mathrm{3}\left[\mathrm{50}+\mathrm{3}\left(\mathrm{45}\right)\right]=\mathrm{555} \\ $$
Commented by hardmath last updated on 10/May/24
perfect dear professor thankyou
$$\mathrm{perfect}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{thankyou} \\ $$

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