Let and let be integers

MATHEMATICS · Subjective · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Let

b>1

and let

n>m\ge 0

be integers. (i) Show that

b^{n}>b^{m}

and

b^{-n}<b^{-m}

. (ii) Hence prove that the sum

b^{n}+b^{-n}

is greater than the sum

b^{m}+b^{-m}

. (iii) Using part (ii), arrange the numbers

b^{2}+b^{-2}

b^{1}+b^{-1}

and

b^{0}+b^{0}

in increasing order.