Without calculating, determine the following...

Question

You are given two sequences t_n = 4 + (n –1)7 and p_n = 4(2)^n –1. Without calculating, which term is bigger t₅₃₄ or p₅₃₄? Why?

Please answer in an organized and neat way! Thank you :)

Answer 1

check at n=1, t=p

look at function f(n)=4(2)^n –1-4 - (n –1)7

at n=1 f=0

n=2 f=-3

n=3 f=0

n=4 f=7,,,yay it is positive

derivative of f(n) with respect to n

f'(n)=4ln2*2^n-1-7

f'(n)>0 ,,,n>2 ensures f'(n)>0

have n=4 f>0, and for n>2 f'>0

guarantees p_n>t_n when n= or greater than 4.

Answer 2

523rd term of p = about a couple quintillion^2

>> than the 523rd term of t= about 10 quintillion

quintillion = 10^19 = 1,000,000,000,000,000,000

quintillion squared = 1 with 84 zeroes after it = astronomically off the charts more >> quintillion

t_{533 <<<< p534}

_{in math they usually stop with >> or << but in this case a few more would seem appropriate}