In the number analogy “65 : 82 :: 145 : ____”, which option follows the same square-plus-one pattern?

Introduction / Context:
This numerical analogy question requires you to spot a less obvious pattern involving perfect squares and a constant adjustment. The pair “65 : 82” appears at first to be just two unrelated numbers, but a careful look reveals a pattern based on expressing each number in the form a^2 + 1. Once this pattern is recognised, you must apply it to 145 and choose the correct corresponding number from the options. This tests your ability to work with perfect squares and generalise a rule from one example to another.

Given Data / Assumptions:

• First pair: 65 and 82.
• Second starting number: 145.
• Options: 165, 168, 169, 170, 160.
• We assume the relationship uses integer squares with a simple constant added.

Concept / Approach:
Begin by expressing 65 in terms of a nearby perfect square. Notice that 8^2 = 64, so 65 can be written as 64 + 1, that is 8^2 + 1. Now look at 82. Here, 9^2 = 81 and 82 is 81 + 1, or 9^2 + 1. This suggests that the pair is of the form “a^2 + 1 : (a + 1)^2 + 1”. Specifically, 65 = 8^2 + 1 and 82 = 9^2 + 1. Applying this pattern to 145, we try to express 145 as a^2 + 1. Since 12^2 = 144, we get 145 = 12^2 + 1. Therefore, the next number must be (12 + 1)^2 + 1 = 13^2 + 1 = 169 + 1 = 170. Thus, the correct answer is 170.

Step-by-Step Solution:

Step 1: Rewrite 65 in the form a^2 + 1. Since 8^2 = 64, we have 65 = 8^2 + 1. Step 2: Rewrite 82 similarly. Since 9^2 = 81, 82 = 9^2 + 1. Step 3: Notice that moving from 65 to 82 corresponds to moving from 8^2 + 1 to 9^2 + 1, so the pattern is “a^2 + 1 → (a + 1)^2 + 1”. Step 4: Express 145 in the same form. Since 12^2 = 144, 145 = 12^2 + 1. Step 5: Apply the rule: the corresponding number should be (12 + 1)^2 + 1 = 13^2 + 1. Step 6: Compute 13^2 = 169 and then add 1 to get 170.

Verification / Alternative check:
Verify that other options cannot fit the same pattern. If we tried 168, it would equal 12^2 + 24, not 13^2 + 1. The number 169 is itself a perfect square (13^2), but the pattern requires square plus one, not just a square. The numbers 165 and 160 are not of the form 13^2 + 1 either. Only 170 matches the required structure.

Why Other Options Are Wrong:
165, 168, 169, and 160 either do not represent any direct square plus one in the sequence after 12^2 or simply ignore the “+1” part of the pattern. For example, 169 equals 13^2 but not 13^2 + 1, so it breaks the rule established by 65 and 82. The analogy must preserve the exact transformation, which is moving from a^2 + 1 to (a + 1)^2 + 1.

Common Pitfalls:
A typical mistake is to pick 169 simply because it is a square and seems mathematically significant. Another is to attempt a linear increment rule like “add 17” without checking whether it fits the form of both pairs in a consistent way. When you see numbers that are just one more than a perfect square, always try to express them as a^2 + 1 and see whether a chain pattern emerges.

Final Answer:
By using the pattern “a^2 + 1 → (a + 1)^2 + 1”, 65 = 8^2 + 1 maps to 82 = 9^2 + 1, and 145 = 12^2 + 1 maps to 170 = 13^2 + 1. So the correct completion is 170.