Which is the largest?

Algebra Level 1

2 2 1 , 3 2 2 , 4 2 3 , 5 2 4 \large \frac{ 2^2}{1} , \ \frac{ 3^2 }{ 2}, \ \frac{ 4^2 } { 3}, \ \frac{ 5^2 } { 4}

Which of the above is the largest?

4 2 3 \frac{4^2}{3} 2 2 1 \frac{ 2^2 } { 1} 5 2 4 \frac{5^2}{4} 3 2 2 \frac{3^2}{2}

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1 solution

Chew-Seong Cheong
Oct 20, 2016

The general expression for the four terms is a n = ( n + 1 ) 2 n a_n = \dfrac {(n+1)^2}n , where n = 1 , 2 , 3 , 4 n = 1, 2, 3, 4 . Consider the quotient a n + 1 a n \dfrac {a_{n+1}}{a_n} , we have:

a n + 1 a n = ( n + 2 ) 2 n + 1 × n ( n + 1 ) 2 = n 3 + 4 n 2 + 4 n n 3 + 3 n 2 + 3 n + 1 > 1 for n 1 \begin{aligned} \frac {a_{n+1}}{a_n} & = \frac {(n+2)^2}{n+1} \times \frac n{(n+1)^2} \\ & = \frac {n^3+4n^2+4n}{n^3+3n^2+3n+1} > 1 & \small \color{#3D99F6}{\text{for }n \ge 1} \end{aligned}

Therefore, a n a_n increases with n n , so the largest of the four terms is a 4 = 5 2 4 a_4 = \boxed{\dfrac {5^2}4} .

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