Understanding Continued Fractions
The problem asks us to find the product of the integer coefficients \(a, b, c, d, e\) in a given continued fraction.
The expression is given as:
To find the values of \(a, b, c, d, e\), we can use the method of continued fraction expansion, which involves repeatedly taking the reciprocal and finding the integer and fractional parts.
Step-by-Step Calculation of Coefficients
First, let's invert both sides of the equation:
Now, we perform successive divisions to find the integer parts:
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Divide \(972\) by \(421\):
\( \frac{972}{421} = 2 + \frac{130}{421} \)Here, the integer part is \(2\). So, \(a = 2\). The remaining fraction is \(\frac{130}{421}\).
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Now consider the next part of the fraction, which equals \(\frac{130}{421}\). We need to find \(b\). Taking the reciprocal:
\( b + \frac{1}{c + \frac{1}{d + \frac{1}{e}}}} = \frac{421}{130} \)Divide \(421\) by \(130\):
\( \frac{421}{130} = 3 + \frac{31}{130} \)The integer part is \(3\). So, \(b = 3\). The remaining fraction is \(\frac{31}{130}\).
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Consider the next part, which equals \(\frac{31}{130}\). We need to find \(c\). Taking the reciprocal:
\( c + \frac{1}{d + \frac{1}{e}} = \frac{130}{31} \)Divide \(130\) by \(31\):
\( \frac{130}{31} = 4 + \frac{6}{31} \)The integer part is \(4\). So, \(c = 4\). The remaining fraction is \(\frac{6}{31}\).
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Consider the next part, which equals \(\frac{6}{31}\). We need to find \(d\). Taking the reciprocal:
\( d + \frac{1}{e} = \frac{31}{6} \)Divide \(31\) by \(6\):
\( \frac{31}{6} = 5 + \frac{1}{6} \)The integer part is \(5\). So, \(d = 5\). The remaining fraction is \(\frac{1}{6}\).
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Finally, we have the last term:
\( \frac{1}{e} = \frac{1}{6} \)This implies \(e = 6\).
Calculating the Product \(a \times b \times c \times d \times e\)
We have found the integer values:
- \(a = 2\)
- \(b = 3\)
- \(c = 4\)
- \(d = 5\)
- \(e = 6\)
Now, we calculate the product:
Performing the multiplication:
- \(2 \times 3 = 6\)
- \(6 \times 4 = 24\)
- \(24 \times 5 = 120\)
- \(120 \times 6 = 720\)
Therefore, the value of \(a \times b \times c \times d \times e\) is \(720\).
Similar Questions
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Select the answer using the code given below.
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Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Let XYZ be a three-digit number, where (x + y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by
