Square-increment series — find the missing term: Complete the series: 20, 21, 25, 34, 50, ?, 111

Introduction / Context:This sequence adds consecutive perfect squares to generate the next term. Once identified, simply continue the square increments to fill the blank.

Given Data / Assumptions:

• Series: 20, 21, 25, 34, 50, ?, 111
• Differences: +1, +4, +9, +16, +25, +36

Concept / Approach:Recognize that the increments are 1^2, 2^2, 3^2, 4^2, 5^2, 6^2. Insert the 5^2 step to find the missing value, then confirm the final step uses 6^2 to reach 111.

Step-by-Step Solution:50 + 5^2 = 50 + 25 = 7575 + 6^2 = 75 + 36 = 111

Verification / Alternative check:Earlier steps: 20→21 (+1), 21→25 (+4), 25→34 (+9), 34→50 (+16), consistent.

Why Other Options Are Wrong:65 or 70 would break the final sum to 111 using the 6^2 increment.

Common Pitfalls:Using linear differences; here differences themselves follow a square sequence.

Final Answer:75