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Question Number 48795 by rahul 19 last updated on 28/Nov/18

Find the sum:  ^(30) C_0 .5^(100) −^(30) C_1 .5^(98) .3^3 +^(30) C_2 .5^(96) .3^6 − ...=?

$${Find}\:{the}\:{sum}: \\ $$$$\:^{\mathrm{30}} {C}_{\mathrm{0}} .\mathrm{5}^{\mathrm{100}} −^{\mathrm{30}} {C}_{\mathrm{1}} .\mathrm{5}^{\mathrm{98}} .\mathrm{3}^{\mathrm{3}} +^{\mathrm{30}} {C}_{\mathrm{2}} .\mathrm{5}^{\mathrm{96}} .\mathrm{3}^{\mathrm{6}} −\:...=? \\ $$

Commented by rahul 19 last updated on 29/Nov/18

T_r = Σ_(r=0) ^(30) (−1)^(r   30) C_r  .(5)^(−2r+100) .(3)^(3r)   T_r = Σ_(r=0) ^(30) ^(30) C_r (25)^(−r+50) (−27)^r   T_r = (25)^(20)  Σ_(r=0) ^(30) ^(30) C_r (25)^(30−r) (−27)^r   T_r = (25)^(20)  (25−27)^(30)   T_(r ) = 2^(30) 5^(40) .

$${T}_{{r}} =\:\underset{{r}=\mathrm{0}} {\overset{\mathrm{30}} {\sum}}\left(−\mathrm{1}\right)^{{r}\:\:\:\mathrm{30}} {C}_{{r}} \:.\left(\mathrm{5}\right)^{−\mathrm{2}{r}+\mathrm{100}} .\left(\mathrm{3}\right)^{\mathrm{3}{r}} \\ $$$${T}_{{r}} =\:\underset{{r}=\mathrm{0}} {\overset{\mathrm{30}} {\sum}}\:^{\mathrm{30}} {C}_{{r}} \left(\mathrm{25}\right)^{−{r}+\mathrm{50}} \left(−\mathrm{27}\right)^{{r}} \\ $$$${T}_{{r}} =\:\left(\mathrm{25}\right)^{\mathrm{20}} \:\underset{{r}=\mathrm{0}} {\overset{\mathrm{30}} {\sum}}\:^{\mathrm{30}} {C}_{{r}} \left(\mathrm{25}\right)^{\mathrm{30}−{r}} \left(−\mathrm{27}\right)^{{r}} \\ $$$${T}_{{r}} =\:\left(\mathrm{25}\right)^{\mathrm{20}} \:\left(\mathrm{25}−\mathrm{27}\right)^{\mathrm{30}} \\ $$$${T}_{{r}\:} =\:\mathrm{2}^{\mathrm{30}} \mathrm{5}^{\mathrm{40}} . \\ $$

Answered by ajfour last updated on 28/Nov/18

=(25)^(20) [^(30) C_0 (25)^(30) −^(30) C_1 (25)^(29) (27)+..]  = (25)^(20) (25−27)^(30)    = 2^(30) 5^(40)  .

$$=\left(\mathrm{25}\right)^{\mathrm{20}} \left[\:^{\mathrm{30}} {C}_{\mathrm{0}} \left(\mathrm{25}\right)^{\mathrm{30}} −^{\mathrm{30}} {C}_{\mathrm{1}} \left(\mathrm{25}\right)^{\mathrm{29}} \left(\mathrm{27}\right)+..\right] \\ $$$$=\:\left(\mathrm{25}\right)^{\mathrm{20}} \left(\mathrm{25}−\mathrm{27}\right)^{\mathrm{30}} \: \\ $$$$=\:\mathrm{2}^{\mathrm{30}} \mathrm{5}^{\mathrm{40}} \:. \\ $$

Commented by rahul 19 last updated on 29/Nov/18

thank you sir!����

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