What should come in place of ‘?’ in the given series? 17 33 65 129 ? 513

Concept: When a series grows too fast for a constant difference, test whether each term is produced from the previous one by a fixed multiplier plus a fixed constant. The diagnostic is to take successive differences: if those differences themselves form a geometric progression — each one a fixed multiple of the one before it — then such a rule governs the series, and the multiplier of the differences is exactly the multiplier applied to the terms.

Application: the given series is 17, 33, 65, 129, ?, 513.

1. Take the successive differences of the known terms: 33 − 17 = 16, 65 − 33 = 32 and 129 − 65 = 64.

2. These differences are not equal, but they double: 16 × 2 = 32 and 32 × 2 = 64. So the multiplier is 2 and the next difference must be 64 × 2 = 128.

3. Add that difference to the term before the blank: 129 + 128 = 257.

4. Written as a recurrence, the rule is an+1 = 2an − 1, and it reproduces every given term: 2 × 17 − 1 = 33, 2 × 33 − 1 = 65 and 2 × 65 − 1 = 129.

Cross-check: The term after the blank must obey the same rule, and 2 × 257 − 1 = 513, which is exactly the last given term, so the pattern closes on both sides. Each term of the series is also one more than a power of two:

17 = 24 + 1, 33 = 25 + 1, 65 = 26 + 1, 129 = 27 + 1, 257 = 28 + 1, 513 = 29 + 1.

Contrast:

• 260 makes the difference 131 instead of 128, which breaks the doubling of the differences, and it would force the next term to be 2 × 260 − 1 = 519.

• 290 makes the difference 161, so the differences would read 16, 32, 64, 161, and the next term would have to be 2 × 290 − 1 = 579.

• 250 makes the difference 121, which is smaller than the previous difference of 64 doubled, and the next term would have to be 2 × 250 − 1 = 499.

Hence the term that belongs in place of ‘?’ is 257.