1, 5, 32, 288 ?
To solve this sequence problem, we need to determine the pattern that generates the numbers and apply it to find the missing term.
Given sequence:
1, 5, 32, 288, ?
Let's analyze the difference between consecutive terms:
- \(5 - 1 = 4\)
- \(32 - 5 = 27\)
- \(288 - 32 = 256\)
Observing these differences:
- 4 can be expressed as \(2^2\)
- 27 can be expressed as \(3^3\)
- 256 can be expressed as \(4^4\)
We notice that the differences correspond to perfect powers, specifically, the differences between terms are increasing perfect powers of numbers from base 2 upwards.
Thus, the pattern gives each term as the summation of a perfect power based on its position:
- First term: \(1\) (base case)
- Second term: \(1 + 2^2 = 5\) (adds \(4\))
- Third term: \(5 + 3^3 = 32\) (adds \(27\))
- Fourth term: \(32 + 4^4 = 288\) (adds \(256\))
- Fifth term: \(288 + 5^5\)
Calculating the fifth term:
\(5^5 = 3125\)
Thus, the fifth term is:
\(288 + 3125 = 3413\)
This matches the correct option, 3413. Therefore, the missing number that replaces the question mark is 3413.
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