Define the terms

Using the formula for the $n$-th term of an A.P., we write the equations for the 8th and 12th terms.

$a + 7d = 82, \quad a + 11d = 478$

Calculate common difference (d)

Subtract the first equation from the second to isolate $d$.

$(a + 11d) - (a + 7d) = 478 - 82 \implies 4d = 396 \implies d = 99$

Calculate 50th term $(a_50)$

Use the 12th term and the common difference to find the 50th term using the relation $a_{50} = a_{12} + (50-12)d$.

$a_{50} = 478 + 38 \times 99$

Final Calculation

Multiply 38 by 99 and add the result to 478.

$a_{50} = 478 + 3762 = 4240$