1 , 2 , 4 , 13 , 31 , 112 , ?
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213
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224
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729
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none
The sequence follows a pattern where each term is obtained by multiplying the previous term by an increasing integer and adding successive square numbers: 1×2+0²=2, 2×2+1²=5 (not 4), wait. The actual pattern appears to be: multiply by 2 and add squares in sequence 0, 1, 4, 9, 16... So: 1×2+0=2, 2×2+0=4, 4×3+1=13, 13×2+5=31, 31×3+19=112... The pattern gives 112×2+0=224.
To solve this sequence, let's look for a pattern or rule that governs the relationship between the numbers.
Looking at the given sequence: 1, 2, 4, 13, 31, 112, ?
We can observe that each number in the sequence is obtained by manipulating the previous number in some way.
Let's examine the differences between consecutive terms:
2 - 1 = 1 4 - 2 = 2 13 - 4 = 9 31 - 13 = 18 112 - 31 = 81
The differences between consecutive terms are not constant. Therefore, the sequence does not follow a simple arithmetic progression or geometric progression.
Let's try looking at the differences between the differences:
2 - 1 = 1 9 - 2 = 7 18 - 9 = 9 81 - 18 = 63
The second differences are not constant either. So, it doesn't seem to be a quadratic sequence.
However, if we observe closely, we can notice that the sequence is formed by adding the square of a number to the previous term. Let's break it down:
1 + 1^2 = 2 2 + 2^2 = 4 4 + 3^2 = 13 13 + 4^2 = 31 31 + 5^2 = 112
Following this pattern, the next term in the sequence would be:
112 + 6^2 = 112 + 36 = 148
Therefore, the missing term in the sequence is 148. However, none of the given options matches this value.
Hence, the correct answer is D) none.