To solve the problem, we start by analyzing the given recurrence relation (a_{n+1} = 2a_n + 1) and the difference between consecutive terms.

Step 1: Relate consecutive differences

Let (dn = a{n+1} - a_n). Using the recurrence relation:
[dn = a{n+1} - a_n = (2a_n + 1) - a_n = a_n + 1]

Next, express (dn) in terms of (d{n-1}):
[an = 2a{n-1} + 1]
[d_n = an + 1 = (2a{n-1} + 1) + 1 = 2(a{n-1} + 1) = 2d{n-1}]

This means the differences form a geometric sequence with ratio 2.

Step 2: Assume initial term (a_1 = 1) (common in such sequences)

• (a_1 = 1)
• (d_1 = a_2 - a_1 = (2*1 +1) -1 = 2)
• (d_2 = 2*d_1 = 4)
• (d_3 = 2*d_2 = 8)
• (d_4 = 2*d_3 = 16)
• (d_5 = 2*d_4 = 32)

Step 3: (a_6 - a_5 = d_5 = 32)

Answer: (\boxed{32})

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