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/easy to read format. And please only include the information that would be understood by a Grade 11. Thank you :)

Answer 1

t_n is an arithmetic sequence that increases linearly by 7 from one term to the next. p_n is a geometric sequence that doubles from one term to the next. Although doubling may seem a lot smaller than increasing by 7, consider how both change for larger terms. Since we are asked not to "calculate", I'll just compare generally. If t_n is at a term value of say 1000, its next value will be 1007. But if p_n is at a term value of 1000, its next term is 2000 (notice how t_n always increases by 7 but the amount p_n increases gets way larger as the term numbers get bigger, In this example, p_n increased by 993 more that t_n). And what if t_n is at a term value of 1,000,000? Again, its next term is 1,000,007 while the next term for p_n would be 2,000,000, an amount 999,993 more than t_n.

Although t_n may grow faster at the beginning, it will not take long for p_n to catch up and completely blow t_n out of the water.

Answer 2

p534 would be greater

here in the tn series the series initially seems to be greater but after a few values of n the pn series becomes greater. This is because a power series( geometric progression) is greater than a normal arithmetic progression tn.

Answer 3