The easiest and fastest solution is by this way: we must only notice that we deal with geometric series with increasing exponents. Then we must find the sum of that geometric series. The first exponent of the first 2 is 2^(1/2). The second exponent of the first 54 is 54^(1/4). So the first pair (2 and 54) we can rewrite like that: 2^(1/2) * 54^(1/4) = 2^(1/4) * 2^(1/4) *54^(1/4) = (2*2*54)^(1/4) = 216^(1/4) the second pair (2 and 54) we can rewrite like that: 2^(1/8) * 54^(1/16) = 2^(1/16) * 2^(1/16) *54^(1/16) = (2*2*54)^(1/16) = 216^(1/16) the third pair (2 and 54) we can rewrite like that: 2^(1/32) * 54^(1/64) = 2^(1/64) * 2^(1/64) *54^(1/64) = (2*2*54)^(1/64) = 216^(1/64) We have 216^(1/4) *216^(1/16) * 216^(1/64)... and so on For that geometric series a = 1/4 and r = 1/4 and n = 216 The formula: Sum = n^ [a / (1-r)] Sum = 216^(1/4) / (1 -( 1/4)) Sum = 216^ (1/4) : (3/4) = 216^ (1/4) * (4/3) = 216^1/3 216^1/3 = 6
