+199 Let \(a \bowtie b = a+\sqrt{b+\sqrt{b+\sqrt{b+...}}}\) .If \(4\bowtie y = 10 \) ,then find the value of \(y\).
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.
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!
