Question

If $R=\frac{({30}^{65}-{29}^{65})}{({30}^{64}-{29}^{64})}$ , then (CAT 2005)

• A. 0<R<=0.1
• B. 0.1<R<=0.5
• C. 0.5< R
• D. R>1

Answer 1

The correct option is D R>1

$R=\frac{{30}^{65}-{29}^{65}}{{30}^{64}+{29}^{64}}$
$\because a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}{b}^2+.....+b^{n-1})$
$\therefore R=\frac{(30-29)[{30}^{64}+({30}^{63}\times 29)+....+{29}^{64}]}{{30}^{64}+{29}^{64}}$
$\because {30}^{64}+{30}^{63}\times 29+...+{29}^{64}>{30}^{64}+{29}^{64}$
$\therefore R>1$
Hence, option 4.
Alternative approach: Unitary Method

$R=\frac{({30}^{65}-{29}^{65})}{({30}^{64}+{29}^{64})}$
If you take this number to be of the form 'n' and 'n+1' instead of 30 and 29, you can solve it using simple numbers like $1\left(n\right)$ and $2(n+1)$

i.e $\frac{(2^3-1^3)}{(2^2+1^2)}=\frac{7}{5}>1$. Only 1 option satisfies this. Option (d)