1. **State the problem:** Find the number of terms in the arithmetic progression (AP) 6, 9, 12, ..., 78. 2. **Recall the formula for the nth term of an AP:** $$a_n = a_1 + (n-1)d$$ where $a_n$ is the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms. 3. **Identify the known values:** - First term $a_1 = 6$ - Common difference $d = 9 - 6 = 3$ - Last term $a_n = 78$ 4. **Substitute into the formula and solve for $n$:** $$78 = 6 + (n-1) \times 3$$ 5. **Simplify the equation:** $$78 - 6 = (n-1) \times 3$$ $$72 = 3(n-1)$$ 6. **Divide both sides by 3:** $$\frac{72}{3} = n - 1$$ $$24 = n - 1$$ 7. **Add 1 to both sides to find $n$:** $$n = 24 + 1 = 25$$ **Final answer:** The number of terms in the arithmetic progression is **25**.