In a coded arithmetic operation denoted by the symbol α, it is given that 18 α 1 = 34, 19 α 2 = 34 and 18 α 3 = 30. Using the same rule, what is the value of 18 α 8 ?

Introduction / Context:
This question defines a custom binary operation represented by the symbol α. You are given three input output pairs involving 18, 19 and small integers, and you must find the result when 18 operates with 8 under the same rule. This type of reasoning problem tests your ability to infer a formula linking two input numbers to an output and then apply it to a new pair.

Given Data / Assumptions:

• 18 α 1 = 34.
• 19 α 2 = 34.
• 18 α 3 = 30.
• The same functional rule applies to all instances of α.
• We need to compute 18 α 8.

Concept / Approach:
Because the results are close to twice the first operand, it is natural to test expressions involving multiples of the first number and the second number. Observing that 18 α 1 and 19 α 2 both equal 34 suggests that the operation combines the first number and the second number in a symmetric way. One efficient approach is to check a simple linear expression like 2 * first minus k times second and determine whether a single consistent k fits all the examples. Once the rule is discovered, it can be directly applied to 18 α 8.

Step-by-Step Solution:
Step 1: Look at 18 α 1 = 34. Twice the first operand is 2 * 18 = 36. Step 2: To get 34 from 36, we subtract 2, which is 2 * 1. This suggests a pattern 2 * first - 2 * second. Step 3: Test this tentative rule with 19 α 2 = 34. 2 * 19 = 38 and 2 * 2 = 4, so 38 - 4 = 34, which matches the given value. Step 4: Test again with 18 α 3 = 30. 2 * 18 = 36 and 2 * 3 = 6, so 36 - 6 = 30, again matching the given result. Step 5: This confirms that the operation is defined by the formula: first α second = 2 * first - 2 * second. Step 6: Now compute 18 α 8 using this rule. 2 * 18 = 36. 2 * 8 = 16. 36 - 16 = 20.

Verification / Alternative check:
The rule can be written compactly as α(a, b) = 2a - 2b. Applying this to each given pair reproduces the outputs exactly, which makes it much more convincing than a rule that works only for some examples. For 18 and 8, no alternative simple formula is needed, because 2 * 18 - 2 * 8 gives 20 cleanly. Since 20 appears among the answer choices, and the rule is fully consistent, the answer is reliable.

Why Other Options Are Wrong:
Options 6, 16, 8 and 18 would require a different formula, such as 2 * first - second or first - 2 * second, but these do not match all the given examples simultaneously. For instance, 2 * 19 - 2 does not give 34, and 2 * 18 - 3 does not give 30. Because the question assumes one fixed definition of α, only the value that agrees with all examples can be correct. That value is 20.

Common Pitfalls:
A common mistake is to try complex patterns involving squares or digit sums when a simple linear relation is sufficient. Another is to stop once a formula fits one example without checking the others. Always verify a proposed rule against all given pairs; if it fails even once, it must be rejected or modified. Testing a small set of reasonable formulas, such as weighted sums or differences of the operands, usually reveals the correct one quickly.

Final Answer:
Using the rule first α second = 2 * first - 2 * second, the value of 18 α 8 is 20.