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Let \(a \bowtie b = a+\sqrt{b+\sqrt{b+\sqrt{b+...}}}\)  .If \(4\bowtie y = 10 \)  ,then find the value of \(y\).

 Jun 18, 2023
 #1
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To find the value of y in the equation "4 bowtie y = 10," we need to substitute the given equation into the definition of the bowtie operator.

Let's replace b with y in the expression for a bowtie b:

a + sqrt(b + sqrt(b + sqrt(b + ...))) = a + sqrt(y + sqrt(y + sqrt(y + ...)))

Now, we can substitute this expression back into the original equation:

4 bowtie y = 10 4 + sqrt(y + sqrt(y + sqrt(y + ...))) = 10

Let's focus on the nested square root expression:

sqrt(y + sqrt(y + sqrt(y + ...)))

We can observe that this nested square root expression is equivalent to the original expression on the right side, which we know to be 4 bowtie y. Therefore, we can replace the nested square root expression with 4 bowtie y:

sqrt(y + 4 bowtie y) = 10

Squaring both sides of the equation, we have:

y + 4 bowtie y = 100

Since we know that 4 bowtie y is equal to 10, we can substitute that value:

y + 10 = 100

Subtracting 10 from both sides:

y = 100 - 10

y = 90

Therefore, the value of y that satisfies the equation "4 bowtie y = 10" is y = 90.

 Jun 18, 2023
 #2
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Good try but the answer was 30.

 Jun 18, 2023
 #3
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Hi xud33,

We just substitute a=4 and b=y to get:

\(4+\sqrt{y+\sqrt{y+\sqrt{y+...}}} =10\)

Thus, 

\(\sqrt{y+\sqrt{y+\sqrt{y+...}}}=6\)

Next, the idea is to square both sides:

\(y+\sqrt{y+\sqrt{y+...}}=36\)

But notice, these nested squareroots are equal to 6 from the previous line!
Hence,

 \(y+6=36 \\ \iff y=30\)

I hope this helps!

 Jun 18, 2023
 #4
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Thanks for the help!

 Jun 18, 2023

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