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Find the missing number in the series: 7, 15, 31, 63, 127, ?
254
255
256
257
255
Identify the pattern as 2n + 1 (doubling the number and adding 1) or recognize the powers of 2 minus 1 (2ᵏ - 1).
Identify the pattern as 2n + 1 (doubling the number and adding 1) or recognize the powers of 2 minus 1 (2ᵏ - 1).
Analyze the pattern of progression
Observe the transitions: 7 * 2 + 1 = 15; 15 * 2 + 1 = 31; 31 * 2 + 1 = 63; 63 * 2 + 1 = 127.
Calculate the missing term
Apply the same logic to the last known number: 127 * 2 + 1 = 254 + 1 = 255.
Alternative approach (Powers of 2)
The series follows (2³)-1=7, (2⁴)-1=15, (2⁵)-1=31, (2⁶)-1=63, (2⁷)-1=127. The next term is (2⁸)-1, which is 256 - 1 = 255.
A: 254 is the result of 127*2 without adding the increment of 1. C: 256 is 2⁸, missing the subtraction of 1. D: 257 is 256 + 1, failing the logic of the series.
B is correct because the series follows a consistent logic of multiplying the previous term by 2 and adding 1, resulting in 127 * 2 + 1 = 255.
When a series grows rapidly, always check for multiplication factors (x2, x3) or squares/cubes first, as they are the most common patterns in competitive exam number series.
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