If \(a+\displaystyle\frac{1}{a}=2\), then find the value of \(a^{34}+\displaystyle\frac{1}{a^{9}}\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
Clearing the fraction turns the condition into a perfect square equal to zero, so \(a\) has only one value, \(a=1\). Every power of 1 is 1.
Multiplying by \(a\): \(a^2+1=2a\), so \(a^2-2a+1=0\)
\((a-1)^2=0\), so \(a=1\)
\(a^{34}=1\) and \(a^{9}=1\), so \(\displaystyle\frac{1}{a^{9}}=1\)
\(a^{34}+\displaystyle\frac{1}{a^{9}}=1+1=2\)