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Which of the following options should come in place of the question mark (?) in the given series? 4 5 9 18 34?
59
60
54
52
59
Look at the differences between consecutive terms: 1, 4, 9, 16. These are squares of consecutive integers (1 squared, 2 squared, 3 squared, 4 squared). The next difference must be 5 squared (25).
Look at the differences between consecutive terms: 1, 4, 9, 16. These are squares of consecutive integers (1 squared, 2 squared, 3 squared, 4 squared). The next difference must be 5 squared (25).
Analyze the differences between consecutive terms
Find the difference between each adjacent pair: 5 - 4 = 1, 9 - 5 = 4, 18 - 9 = 9, 34 - 18 = 16.
Identify the pattern in the differences
The differences are 1, 4, 9, 16, which correspond to the squares of natural numbers: 1 squared, 2 squared, 3 squared, and 4 squared.
Calculate the missing number
Following the sequence, the next difference must be 5 squared, which is 25. Adding this to the last term gives 34 + 25 = 59.
B: 60 is incorrect because it adds 26 instead of 25. C: 54 is incorrect because it adds 20 instead of 25. D: 52 is incorrect because it adds 18 instead of 25.
A is correct because the series follows an increasing difference pattern of consecutive squares (1 squared, 2 squared, 3 squared, 4 squared, 5 squared), yielding 34 + 25 = 59.
Whenever differences in a series grow rapidly as perfect squares (1, 4, 9, 16, 25), always check the sequence of squares of consecutive integers.
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