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rshm15varma

(1+1/3) (1+1/3^2) (1+3^3) (1+3^3).........(1+3^2n)=3/2(1-(3)^(-2)^(n+1))

nadeemoidu

There is a mistake in your question . the 3rd term is 3^ 4 and fourth term is 3^8 the general term is 3^2^n not 3^n

Let S=( 1+ 1/3 ) ( 1+ 1/3 ²) ( 1+ 1/3⁴ ) ( 1+ 1/ 3⁸) .....( 1+ 1/3 ^{2^n})

( 1- 1/3) S =( 1-1/3) ( 1+ 1/3 ) ( 1+ 1/3² ) ( 1+ 1/3 ⁴) ( 1+ 1/ 3⁸ ) .....( 1+ 1/3 ^{2^n})

( 2/3) S = ( 1- 1/3² ) ( 1+ 1/3² ) ( 1+ 1/3 ⁴) ( 1+ 1/ 3⁸ ) .....( 1+ 1/3 ^{2^n})

= ( 1- 1/3 ⁴)( 1+ 1/3 ⁴) ( 1+ 1/ 3⁸ ) .....( 1+ 1/3 ^{2^n})

= ( 1- 1/ 3⁸ ) ( 1+ 1/ 3⁸ ).....( 1+ 1/3 ^{2^n})

and so on....

= ( 1- 1/3 ^{2^n})( 1+ 1/3 ^{2^n})
=( 1- 1/3 ^{2^n+1})

S= (3/2)( 1- 1/3 ^{2^n+1})

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