#MathUnusual | Find the limit n tends infinity \sqrt2\sqrt2\sqrt2\sqrt2...(n times)

Hello friends, this is Mahesh Yoylekar and you are watching Next Gen Maths. If x1 equal to square root 2, xn plus 1 equal to square root of 2xn and greater than equal to 1 then find the limit n tends to infinity log xn base b. If you want to solve then stop the video otherwise let's start. So we are given xn plus 1 equal to square root 2xn. If you begin n equal to 1 then we have x2 equal to square root of 2x1. So it is square root of 2 into square root of 2. x3 is equal to square root of 2x2. So square root of 2 into square root of 2 into square root of 2. For x4 it is square root of 2x3. So it is square root of 2, square root of 2, square root of 2, square root of 2, 4 times. for xn is equal to square root of 2 xn minus 1 it is square root of 2 square root of 2 square root of 2 up to n times but this form of xn will not help to find the limit n tends to infinity log xn base therefore we write all the terms for x2 it is square root 2 into square root x1 so it is square root 2 into square root of square root 2 the 2 power 1 by 2 and here 2 power 1 by 2 into 1 by 2 so it is 2 power 1 by 4 so 2 power 1 by 2 plus 1 by 4 equal to 2 power 3 by x3 is equal to square 2 into square root of x2 the square root 2 is equal to 2 power 1 by 2 and square root of x2 is equal to 2 power 3 by 4 into 1 by 2 so it is 2 power 3 by 8 therefore 2 power 1 by 2 plus 3 by 8 is equal to 2 power 7 by 8 now x4 is equal to square root 2 into square root of x3 so square root 2 equal to 2 power 1 by 2 and square root of x3 is equal to 2 power 7 by 8 into 1 by 2 it is 2 power 7 by 16 1 by 2 plus 7 by 16 is equal to 2 power 15 by 16 if you observe all the terms then in the first term x2 is equal to 2 power 3 by 4 which is 2 power 4 minus 1 divided by 4 but 4 is equal to 2 power 2 so you can write down 2 power 2 power 2 minus 1 divided by 2 power 2 x3 is equal to 2 power 8 minus 1 divided by 8 but 8 is equal to 2 power 3 it is 2 power 2 power 3 minus 1 divided by 2 power 3 similarly for x4 it is 2 power 2 power 4 minus 1 divided by 2 power 4 therefore xn is equal to 2 power 2 power n minus 1 divided by 2 power now we are going to find the value of limit n tends to infinity log xn base limit n tends to infinity log square root 2 square root 2 up to n times another form limit n tends to infinity log 2 power 2 power n minus 1 divided by 2 power n is equal to limit n tends to infinity by logarithmic property this 2 power n minus 1 divided by 2 power n into log 2 base e so log 2 base is constant so take out from the limit and we have limit n tends to infinity this number can be rewrite in the form of 1 minus 1 upon 2 power n as n tends to infinity 2 power n approach to infinity so 1 upon infinity will be 0 equal to log 2 base e into 1 minus 0 so value is log 2 base e so this example has value log 2 base e thanks for watching for more videos like share subscribe press the bell icon of next gen