A simple game

Algebra Level 5

The numbers 2,4,8,16,...,\( 2^n \) are written on a chalkboard. A student selects two numbers a and b , erase them and replaces them by their average say \( \frac{(a+b)}{2} \) . She performs this operation (n-1) times until only one number is left. Let M denotes the maximum value of this final number. If we determine the formula for M in terms of n. Then we get a pattern like this \( \frac{2^a + 2^b}{3} \) . Where a and b are linear polynomials in variable n. The sum of a and b gives a integer. This integer equals to :

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