Number series – compute the next term: 2, 5, 10, 19, 36, ?

Introduction / Context:
Some series grow by adding a changing difference that follows its own recurrence. Here, the term-to-term increments themselves follow a simple relation, allowing the next term to be predicted cleanly.

Given Data / Assumptions:

• Series: 2, 5, 10, 19, 36, ?
• We examine first differences and look for a recurrence on the differences.

Concept / Approach:
Compute consecutive differences, then see if those differences follow a compact rule (e.g., doubling and subtracting a constant). This “recurrence on differences” is a common pattern in competitive exams.

Step-by-Step Solution:
Differences: 5−2 = 3, 10−5 = 5, 19−10 = 9, 36−19 = 17.Observe the meta-pattern on differences: 3 → 5 → 9 → 17. Each new difference = 2 * (previous difference) − 1.Check: 3*2−1 = 5; 5*2−1 = 9; 9*2−1 = 17. Next difference = 17*2−1 = 33.Next term = 36 + 33 = 69.

Verification / Alternative check:
Attempting multiplicative rules on the original terms leads to inconsistency; the difference-based recurrence fits all transitions exactly and predicts a unique next value of 69.

Why Other Options Are Wrong:
70/71/68 would require next differences of 34/35/32, which break the 2*d − 1 recurrence produced by the established run of differences.

Common Pitfalls:
Mistaking the differences as primes or squares; while 3,5 are prime, the governing rule is the 2*d − 1 recurrence, not primality.

Final Answer:
69