# 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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