If \(a+\displaystyle\frac{1}{a+1}=10\), then find the value of \((a+1)^3+\displaystyle\frac{1}{(a+1)^3}\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1298
Adding 1 to both sides makes the left side \(t+\frac{1}{t}\) with \(t=a+1\). The cube then follows from \(t^3+\frac{1}{t^3}=\left(t+\frac{1}{t}\right)^3-3\left(t+\frac{1}{t}\right)\).
\((a+1)+\displaystyle\frac{1}{a+1}=10+1=11\)
\((a+1)^3+\displaystyle\frac{1}{(a+1)^3}=11^3-3\times11\)
\(=1331-33=1298\)