UPSC CDS I 2024 Math Paper

Concept:Convert the given continued fraction into an ordinary fraction by repeated reciprocal and integer part extraction. The integers obtained become the values of $a, b, c, d, e$ in order.Explanation:Start with the given equation: $\frac{1}{a+\frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}}}} = \frac{421}{972}$.Take reciprocal of both sides: $a+\frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}}} = \frac{972}{421}$.Write $\frac{972}{421}$ as $2 + \frac{130}{421}$. So $a = 2$.Now $\frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}}} = \frac{130}{421}$. Reciprocal: $b+\frac{1}{c+\frac{1}{d+\frac{1}{e}}} = \frac{421}{130}$.Write $\frac{421}{130}$ as $3 + \frac{31}{130}$. So $b = 3$.Then $\frac{1}{c+\frac{1}{d+\frac{1}{e}}} = \frac{31}{130}$. Reciprocal: $c+\frac{1}{d+\frac{1}{e}} = \frac{130}{31} = 4 + \frac{6}{31}$. So $c = 4$.Next $\frac{1}{d+\frac{1}{e}} = \frac{6}{31}$. Reciprocal: $d+\frac{1}{e} = \frac{31}{6} = 5 + \frac{1}{6}$. So $d = 5$.Finally $\frac{1}{e} = \frac{1}{6}$, so $e = 6$.Multiply: $a \times b \times c \times d \times e = 2 \times 3 \times 4 \times 5 \times 6 = 720$.Answer:$720$ (Option A)