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Question-191024




Question Number 191024 by Rupesh123 last updated on 16/Apr/23
Answered by BaliramKumar last updated on 16/Apr/23
3600 = 36×100 = 2^2 ×3^2 ×2^2 ×5^2                2^4 ×3^2 ×5^2                  [4^2 ×9^1 ×25^1 ]   Sum = (((2^5 −1)/(2−1)))(((3^3 −1)/(3−1)))(((5^3 −1)/(5−1))) − (((4^3 −1)/(4−1)))(((9^2 −1)/(9−1)))(((25^2 −1)/(25−1)))                        = 31×13×31 − 21×10×26              = 13(961− 420) = 13×541              = 7033
$$\mathrm{3600}\:=\:\mathrm{36}×\mathrm{100}\:=\:\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{2}} ×\mathrm{5}^{\mathrm{2}} \: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{4}^{\mathrm{2}} ×\mathrm{9}^{\mathrm{1}} ×\mathrm{25}^{\mathrm{1}} \right] \\ $$$$\:\mathrm{Sum}\:=\:\left(\frac{\mathrm{2}^{\mathrm{5}} −\mathrm{1}}{\mathrm{2}−\mathrm{1}}\right)\left(\frac{\mathrm{3}^{\mathrm{3}} −\mathrm{1}}{\mathrm{3}−\mathrm{1}}\right)\left(\frac{\mathrm{5}^{\mathrm{3}} −\mathrm{1}}{\mathrm{5}−\mathrm{1}}\right)\:−\:\left(\frac{\mathrm{4}^{\mathrm{3}} −\mathrm{1}}{\mathrm{4}−\mathrm{1}}\right)\left(\frac{\mathrm{9}^{\mathrm{2}} −\mathrm{1}}{\mathrm{9}−\mathrm{1}}\right)\left(\frac{\mathrm{25}^{\mathrm{2}} −\mathrm{1}}{\mathrm{25}−\mathrm{1}}\right)\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{31}×\mathrm{13}×\mathrm{31}\:−\:\mathrm{21}×\mathrm{10}×\mathrm{26} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{13}\left(\mathrm{961}−\:\mathrm{420}\right)\:=\:\mathrm{13}×\mathrm{541} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{7033} \\ $$

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