Question

if 29/12 can be expressed as a + 1/(b+(1/c+(1/d))) then find value of a^3+b^3+c^3+d^3

Answer 1

Expressing a fraction

Given: $\frac{29}{12}$ can be expressed as $\displaystyle a+\frac{1}{b+\frac{1}{c+\frac{1}{d}}}$

To find: the value of $a^{3}+b^{3}+c^{3}+d^{3}$

Solution:

The fraction is $\displaystyle\frac{29}{12}=2+\frac{5}{12}$

Now, $\displaystyle\frac{1}{b+\frac{1}{c+\frac{1}{d}}}$

$=\displaystyle\frac{1}{b+\frac{d}{cd+1}}$

$=\frac{cd+1}{bcd+b+d}$ which is equal to $\frac{5}{12}$

In numerator, we have $5$, which can be expressed as $2\times 2 + 1$

This gives: $c = d = 2$

The value of $b$ must be $2$ also, in order to get the denominator $(bcd + b + d) = 12$.

We have found: $a = b = c = d = 2$.

• $\displaystyle\frac{29}{12}=2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$

Now, $a^{3}+b^{3}+c^{3}+d^{3}$

$=4a^{3}$, since $a = b = c = d$

$=4\times 2^{3}$

$=4\times 8$

$=\bold{32}$

Answer: $a^{3}+b^{3}+c^{3}+d^{3}=32$.

Answer 2

Answer is given in the attachment.

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