1. **State the problem:** We are given an arithmetic progression (A.P.) with first term $U_1 = -26$, common difference $d = 2$, and the sum of the first $n$ terms $S_n = 324$. We need to find the number of terms $n$. 2. **Recall the formula for the sum of the first $n$ terms of an A.P.:** $$ S_n = \frac{n}{2} [2U_1 + (n-1)d] $$ 3. **Substitute the known values:** $$ 324 = \frac{n}{2} [2(-26) + (n-1)2] $$ 4. **Simplify inside the bracket:** $$ 324 = \frac{n}{2} [-52 + 2(n-1)] $$ $$ 324 = \frac{n}{2} [-52 + 2n - 2] $$ $$ 324 = \frac{n}{2} [2n - 54] $$ 5. **Multiply both sides by 2 to clear the denominator:** $$ 648 = n(2n - 54) $$ 6. **Expand the right side:** $$ 648 = 2n^2 - 54n $$ 7. **Bring all terms to one side to form a quadratic equation:** $$ 2n^2 - 54n - 648 = 0 $$ 8. **Divide the entire equation by 2 to simplify:** $$ n^2 - 27n - 324 = 0 $$ 9. **Solve the quadratic equation using the quadratic formula:** $$ n = \frac{27 \pm \sqrt{(-27)^2 - 4(1)(-324)}}{2} $$ $$ n = \frac{27 \pm \sqrt{729 + 1296}}{2} $$ $$ n = \frac{27 \pm \sqrt{2025}}{2} $$ $$ n = \frac{27 \pm 45}{2} $$ 10. **Calculate the two possible values for $n$:** - $$ n = \frac{27 + 45}{2} = \frac{72}{2} = 36 $$ - $$ n = \frac{27 - 45}{2} = \frac{-18}{2} = -9 $$ 11. **Interpret the results:** Since the number of terms $n$ cannot be negative, we discard $n = -9$. **Final answer:** $$ \boxed{36} $$ terms.