Step 1: Understanding the Concept:
We need to identify the pattern or rule governing the given sequence of numbers to find the missing term. Sequence problems often involve arithmetic or geometric progressions, powers, or combinations of operations.
Step 2: Key Formula or Approach:
Let's analyze the given terms to see if they fit a common pattern. The numbers are 0, 15, 80, ?, 624. These numbers seem to be close to powers of integers.
- 0
- 15 is close to \(16 = 4^2 = 2^4\)
- 80 is close to \(81 = 9^2 = 3^4\)
- 624 is close to \(625 = 25^2 = 5^4\)
Let's test the pattern \(n^4 - 1\).
Or maybe a pattern like \(n^2-1\)?
- For 15: \(4^2 - 1 = 15\).
- For 80: \(9^2 - 1 = 80\).
- For 624: \(25^2 - 1 = 624\).
- For 0: \(1^2 - 1 = 0\).
The bases of the squares are 1, 4, 9, ?, 25. These are themselves perfect squares: \(1^2, 2^2, 3^2, ?, 5^2\).
So the missing base should be \(4^2=16\).
Step 3: Detailed Explanation:
Let the terms of the sequence be denoted by \(a_n\).
The given terms are:
- \(a_1 = 0\)
- \(a_2 = 15\)
- \(a_3 = 80\)
- \(a_4 = ?\)
- \(a_5 = 624\)
Let's examine the pattern observed in Step 2. The terms seem to be of the form \(k^2-1\).
- \(a_1 = 0 = 1^2 - 1\)
- \(a_2 = 15 = 4^2 - 1\)
- \(a_3 = 80 = 9^2 - 1\)
- \(a_5 = 624 = 25^2 - 1\)
The numbers being squared are 1, 4, 9, ?, 25.
Let's look at this sequence of bases: 1, 4, 9, 25.
These are the squares of consecutive integers:
- \(1 = 1^2\)
- \(4 = 2^2\)
- \(9 = 3^2\)
- \(25 = 5^2\)
So the sequence of bases is \(n^2\) for \(n=1, 2, 3, 4, 5\).
The general term of the main sequence is \(a_n = ((n^2)^2) - 1 = n^4 - 1\). Let's test this pattern.
- \(a_1 = 1^4 - 1 = 0\). Correct.
- \(a_2 = 2^4 - 1 = 16 - 1 = 15\). Correct.
- \(a_3 = 3^4 - 1 = 81 - 1 = 80\). Correct.
- \(a_5 = 5^4 - 1 = 625 - 1 = 624\). Correct.
The pattern holds. Now we can find the missing fourth term, \(a_4\):
\[ a_4 = 4^4 - 1 = 256 - 1 = 255 \]
Step 4: Final Answer:
The missing number in the sequence is 255, which corresponds to option (D).