The value of the infinite product `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)

1.	The value of the infinite product `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...`
Answer» `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...` `=6^(1/2+2/4+3/8+4/16)` Now, we will solve `(1/2+2/4+3/8+4/16+...)`. Let `S = 1/2+2/4+3/8+4/16+...` `S = 1/2+2/2^2+3/2^3+...->(1)` `S/2 = 1/2^2+2/2^3+3/2^4+...->(2)` Subtracting `(1) - (2)`, `S/2 = 1/2+1/2^2+1/2^3+...` `=>S/2 = (1/2)/(1-1/2) = 1` `=> S = 2` `:. 6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx... = 6^S = 6^2 = 36`

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The value of the infinite product `6^(1/2)xx6^(2/4)xx6^(3/8)xx6^(4/16)xx...`

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