3 $ 4 Pattern Analysis

The problem presents a mathematical operation '$' defined by two examples:

• 3 $ 4 = 37
• 4 $ 5 = 61

We need to find the value of 5 $ 6 by identifying the pattern.

Rule Derivation

Let the operation be represented as $a \text{ \$ } b$. We test a common pattern involving squares and linear terms: $a \text{ \$ } b = a^2 + b^2 + k \cdot a + m$.

1. Using the first example (3 $ 4 = 37$, so $a=3, b=4$):

   $3^2 + 4^2 + k \cdot 3 + m = 37$ $9 + 16 + 3k + m = 37$ $25 + 3k + m = 37$ $3k + m = 12$ (Equation 1)

2. Using the second example (4 $ 5 = 61$, so $a=4, b=5$):

   $4^2 + 5^2 + k \cdot 4 + m = 61$ $16 + 25 + 4k + m = 61$ $41 + 4k + m = 61$ $4k + m = 20$ (Equation 2)

3. Solve the system of linear equations (1) and (2) for $k$ and $m$.

   Subtracting Equation 1 from Equation 2 gives: $(4k + m) - (3k + m) = 20 - 12$ $k = 8$ Substitute $k=8$ into Equation 1: $3(8) + m = 12 \implies 24 + m = 12 \implies m = -12$.

4. The derived rule is $a \text{ \$ } b = a^2 + b^2 + 8a - 12$.

5 $ 6 Calculation

Apply the rule $a \text{ \$ } b = a^2 + b^2 + 8a - 12$ to find 5 $ 6, using $a=5$ and $b=6$.

The result is 89.