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Question Number 24704 by math solver last updated on 25/Nov/17

Commented by math solver last updated on 25/Nov/17

sum of series upto ′n′ terms is ?

$$\mathrm{sum}\:\mathrm{of}\:\mathrm{series}\:\mathrm{upto}\:'\mathrm{n}'\:\mathrm{terms}\:\mathrm{is}\:? \\ $$

Commented by prakash jain last updated on 25/Nov/17

T_i =(.5)^3 ×(i+1)^3   S_n =Σ_(i=1) ^n (.5)^3 ×(i+1)^3   =(.5)^3 [Σ_(i=0) ^(n+1) i^3 −1]  =(.5)^3 [{(((n+1)(n+2))/2)}^2 −1]

$${T}_{{i}} =\left(.\mathrm{5}\right)^{\mathrm{3}} ×\left({i}+\mathrm{1}\right)^{\mathrm{3}} \\ $$$${S}_{{n}} =\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(.\mathrm{5}\right)^{\mathrm{3}} ×\left({i}+\mathrm{1}\right)^{\mathrm{3}} \\ $$$$=\left(.\mathrm{5}\right)^{\mathrm{3}} \left[\underset{{i}=\mathrm{0}} {\overset{{n}+\mathrm{1}} {\sum}}{i}^{\mathrm{3}} −\mathrm{1}\right] \\ $$$$=\left(.\mathrm{5}\right)^{\mathrm{3}} \left[\left\{\frac{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}{\mathrm{2}}\right\}^{\mathrm{2}} −\mathrm{1}\right] \\ $$

Commented by math solver last updated on 25/Nov/17

gracias.

$$\mathrm{gracias}. \\ $$

Commented by math solver last updated on 25/Nov/17

and how you write the 3rd step?

$$\mathrm{and}\:\mathrm{how}\:\mathrm{you}\:\mathrm{write}\:\mathrm{the}\:\mathrm{3rd}\:\mathrm{step}? \\ $$

Commented by jota+ last updated on 25/Nov/17

{i+1}={2  3  4  5 ...}  {j}      ={1   2  3  4...}    Σ_(j=1) ^(n+1) a_j −a(1)=Σ_(i=1) ^n a_i   (Propiedad telescopica)

$$\left\{{i}+\mathrm{1}\right\}=\left\{\mathrm{2}\:\:\mathrm{3}\:\:\mathrm{4}\:\:\mathrm{5}\:...\right\} \\ $$$$\left\{{j}\right\}\:\:\:\:\:\:=\left\{\mathrm{1}\:\:\:\mathrm{2}\:\:\mathrm{3}\:\:\mathrm{4}...\right\} \\ $$$$ \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{n}+\mathrm{1}} {\sum}}{a}_{{j}} −{a}\left(\mathrm{1}\right)=\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{a}_{{i}} \\ $$$$\left({Propiedad}\:{telescopica}\right) \\ $$

Answered by jota+ last updated on 25/Nov/17

T_i =B_i ^3 =(0.5+0.5i)^3 =  =0.5^3 (1+i)^3

$${T}_{{i}} ={B}_{{i}} ^{\mathrm{3}} =\left(\mathrm{0}.\mathrm{5}+\mathrm{0}.\mathrm{5}{i}\right)^{\mathrm{3}} = \\ $$$$=\mathrm{0}.\mathrm{5}^{\mathrm{3}} \left(\mathrm{1}+{i}\right)^{\mathrm{3}} \\ $$

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