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Let S be infinite sum of the series 2+3 x+4 ^2 x+5 ^3 x+ , where x satisfies the equation |5 x+4|+|5 x-2|=6 . If the least value of S is equal to ( a b ) where a and b are co-prime numbers, then find the value of (a+b) .

Correct answer

151

Step-by-step solution

S=2+3 x+4 ^2 x+ aligned & array l S x=2 x+3 ^2 x+ (1- x)=2+ x+ ^2 x+ array & =1+ 1 1- x & S= 1 1- x + 1 (1+ x)^2 & |5 x+4|+|5 x-2|=6 & x [- 4 5 , 2 5 ] & aligned For the least value of S, x=- 4 5 aligned & . S |_ least = 1 1+ 4 5 + 1 (1+ 4 5 )^2 = 5 9 + 25 81 = 70 81 = a b & a+b=151 aligned

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