1. **State the problem:** We need to solve the expression $$1 + \frac{4}{6} + \frac{4.5}{6.9} + \frac{4.5.6}{6.9.12} + \ldots$$ 2. **Understand the pattern:** The terms appear to be fractions with products in numerator and denominator increasing by one factor each time. 3. **Rewrite terms:** - First term: $1$ - Second term: $\frac{4}{6}$ - Third term: $\frac{4 \times 5}{6 \times 9}$ (assuming $4.5$ means $4 \times 5$ and $6.9$ means $6 \times 9$) - Fourth term: $\frac{4 \times 5 \times 6}{6 \times 9 \times 12}$ 4. **Simplify each term:** - $\frac{4}{6} = \frac{2 \times 2}{2 \times 3} = \frac{2}{3}$ - $\frac{4 \times 5}{6 \times 9} = \frac{20}{54} = \frac{10}{27}$ - $\frac{4 \times 5 \times 6}{6 \times 9 \times 12} = \frac{120}{648} = \frac{20}{108} = \frac{5}{27}$ 5. **Sum the known terms:** $$1 + \frac{2}{3} + \frac{10}{27} + \frac{5}{27} = 1 + \frac{2}{3} + \frac{15}{27}$$ 6. **Convert all to a common denominator 27:** - $1 = \frac{27}{27}$ - $\frac{2}{3} = \frac{18}{27}$ - $\frac{15}{27}$ stays the same 7. **Add:** $$\frac{27}{27} + \frac{18}{27} + \frac{15}{27} = \frac{60}{27} = \frac{20}{9} \approx 2.222...$$ 8. **Conclusion:** The sum of the first four terms is $\frac{20}{9}$ or approximately 2.222. The pattern suggests a series but without more terms or a general formula, we sum only these.