In this number analogy, “50 is to 65 as 122 is to ______”. Choose the number that completes the analogy by following the same perfect square based pattern on both sides.

Introduction / Context:
This number analogy involves recognising a pattern based on nearby perfect squares. The pair “50 : 65” is not linked by a simple addition like plus fifteen alone in a random way. Instead, both 50 and 65 can be related to consecutive perfect squares. You must discover this hidden connection and apply the same idea to the pair “122 : ?” to find the correct number from the options. Such questions test your familiarity with perfect squares and your ability to see how numbers sit between them.

Given Data / Assumptions:

• First pair: 50 corresponds to 65. • Second pair: 122 corresponds to an unknown number. • Options: 157, 145, 147, 155. • We recall important perfect squares: 7² = 49, 8² = 64, 11² = 121, 12² = 144, and so on.

Concept / Approach:
Notice that 50 is one more than 49 and 65 is one more than 64. Here, 49 = 7² and 64 = 8². So the first pair can be interpreted as “one more than a perfect square of 7” mapping to “one more than the next perfect square of 8”. Therefore, we look at 122 in the same way. It is one more than 121, which is 11². Following the pattern, we need the number that is one more than the next square after 121, which is 144 (12²). One more than 144 is 145.

Step-by-Step Solution:
Step 1: Express 50 and 65 in terms of squares. 50 = 49 + 1 = 7² + 1. 65 = 64 + 1 = 8² + 1. Step 2: Identify the pattern. The mapping goes from (7² + 1) to (8² + 1). In other words, from “one more than 7²” to “one more than the next square 8²”. Step 3: Express 122 similarly. 122 = 121 + 1 = 11² + 1. Step 4: Find the next square after 11². The next square is 12² = 144. Step 5: Add one to that square. 144 + 1 = 145. Step 6: Therefore, 122 should correspond to 145 to keep the same pattern “square plus one with increasing base”.

Verification / Alternative check:
We can summarise the rule as: if the first number is n² + 1, the second number should be (n + 1)² + 1. For 50, n = 7, and the partner is (7 + 1)² + 1 = 8² + 1 = 64 + 1 = 65. For 122, n = 11, and the partner is (11 + 1)² + 1 = 12² + 1 = 144 + 1 = 145. No other option fits so neatly into this pattern, confirming that 145 is the correct answer.

Why Other Options Are Wrong:
• 157, 147, 155: None of these numbers can be written as “next perfect square plus one” when the first number is 11² + 1. They do not maintain the consistent square based relationship seen in the 50 : 65 pair.

Common Pitfalls:
Students often try linear operations first, such as adding the same constant or multiplying by a fixed number. While 65 is indeed 50 + 15, that does not extend cleanly to 122 with any of the given options. The presence of numbers close to known squares is a strong hint to think in terms of squares instead of simple differences, which leads directly to the correct pattern.

Final Answer:
The number that correctly completes the analogy is 145.