If 2 @ 3 = 25 and 3 @ 4 = 49, what is 4 @ 5?

Correct — A. The given results, 25 and 49, are perfect squares.

Rule: a @ b = (a + b)².

2 @ 3 = (2 + 3)² = 5² = 25, as given
3 @ 4 = (3 + 4)² = 7² = 49, as given
4 @ 5 = (4 + 5)² = 9² = 81

That is option (a).

Why the others are wrong

• (b)
  100 — 100 = 10² would need a + b = 10, but 4 + 5 = 9. A rule like (a + b + 1)² reaches 100, yet it gives 36 for 2 @ 3, not 25.
• (c)
  121 — 121 = 11² = (5 + 6)², the value the rule gives for 5 @ 6. It is one step further along than 4 @ 5.
• (d)
  91 — 91 is not a perfect square (9² = 81, 10² = 100), so no rule that squares a sum can produce it.

An operator item shows a hidden formula through examples. When the results are perfect squares, try squaring a simple combination of a and b.

Here 25 = 5² and 49 = 7², and 5 and 7 are the sums 2 + 3 and 3 + 4. Test the rule on both examples before you apply it.

Key facts

• (a + b)² turns 2 @ 3 into 25, 3 @ 4 into 49 and 4 @ 5 into 81.
• 25, 49 and 81 are the squares of the odd numbers 5, 7 and 9.
• When b = a + 1, (a + b)² equals 4ab + 1.

Common traps

• Confusing (a + b)² with a² + b²: for 2 @ 3 the first gives 25, the second 13.
• Continuing 25, 49 with the next square, 64: the rule gives 5², 7², 9², so the next value is 81.

No directly related past PYQ was found.