Mr. Bean chooses a number and he keeps on doubling the number followed by…
2026
Mr. Bean chooses a number and he keeps on doubling the number followed by subtracting one from it, if he chooses 3 as initial number and he repeats the operation for 30 times then what is the final result?
- A.
2^31 + 1
- B.
+1
- C.
(2^30) - 2
- D.
(2^30) - 1
Attempted by 263 students.
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Correct answer: A
Answer: 2^31 + 1
Let a_n be the number after n operations, with the initial value a_0 = 3. Each operation applies the rule a_{n+1} = 2 a_n - 1.
Compute the first few terms to observe a pattern: a_1 = 2*3 - 1 = 5 = 2^2 + 1; a_2 = 2*5 - 1 = 9 = 2^3 + 1; a_3 = 17 = 2^4 + 1.
Conjecture the closed form a_n = 2^{n+1} + 1. Verify by induction: if a_n = 2^{n+1} + 1 then
a_{n+1} = 2 a_n - 1 = 2(2^{n+1} + 1) - 1 = 2^{n+2} + 1, so the formula holds for the next step.
Therefore, after 30 operations (n = 30) we have a_30 = 2^{31} + 1.
Final result: 2^31 + 1