To calculate the number of palindromes of  length(2n), consider the following diagram,

which shows that there are as many palindromes of length 2n as there are the strings of length n i.e. the required number of palindromes are 2ⁿ.

To calculate the number of palindromes of length (2n-1) with ‘a’ as the middle letter, consider the following diagram,

which shows that there are as many palindromes of length 2n-1 as there are the strings of length n-1 i.e. the required number of palindromes are 2^n-1.

Similarly the number of palindromes of length 2n-1, with ‘ b ’ as middle letter, will be 2^n-1 as well. Hence the total number of palindromes of length 2n-1 will be 2^n-1 + 2^n-1 = 2 (2^n-1)= 2ⁿ .

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