Number series — differences increase by a constant amount: Complete the series: 39, 52, 78, 117, 169, ?

Introduction / Context:Many series are built by adding differences that themselves form an arithmetic progression. Detecting the pattern in the gaps between consecutive terms is often faster than manipulating the terms directly.

Given Data / Assumptions:

• Series: 39, 52, 78, 117, 169, ?
• Consecutive differences appear to grow regularly.

Concept / Approach:Compute the consecutive differences and examine whether those differences increase by a constant. If so, extend the same rule to get the next term.

Step-by-Step Solution:Differences: 52−39 = 1378−52 = 26 (13 more)117−78 = 39 (again +13)169−117 = 52 (again +13)Next difference = 52 + 13 = 65Next term = 169 + 65 = 234

Verification / Alternative check:The pattern of differences is 13, 26, 39, 52, 65 which is an arithmetic progression with common difference 13.

Why Other Options Are Wrong:246 and 256 would correspond to adding 77 or 87, breaking the +13 increment rule; 182 is too small.

Common Pitfalls:Jumping to geometric reasoning; here the additive structure is clearer.

Final Answer:234