Question

. For any integer n ≥ 2, let A(n) denote the number of ways to fully...

. For any integer n ≥ 2, let A(n) denote the number of ways to fully parenthesize a sum of n terms such as a1 + · · · + an. Examples:

• A(2) = 1, since the only way to fully parenthesize a1 + a2 is (a1 + a2).

• A(3) = 2, since the only ways to fully parenthesize a1 + a2 + a3 are ((a1 + a2) + a3) and
(a1 + (a2 + a3)).

• A(4) = 5, since a1 + a2 + a3 + a4 can be parenthesized as ((a1 + a2) + (a3 +
a4)), (((a1 + a2) + a3) + a4), (a1 + (a2 + (a3 + a4))), ((a1 + (a2 + a3)) + a4), and (a1 +
((a2 + a3) + a4)).

Prove using complete induction that A(n) <= (n 1)n−1 for all integers

n >= 2. Your proof does not need to be formal, but it should have a clear structure and be easy to understand.

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Answer #1

There is an error in the question in the expression A(n)

I have provided the correct expression and solved it by induction

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