Question

Consider the sequence t0 = 3; t1 = 3^3; t2 = 3^33 ; t3 = 3^333
; : : : defined by t0 = 3 and
tn+1 = 3^tn for n 0.

What are the last two digits in t3 = 3333
?

Can you say what the
last three digits are?

Show that the last 10 digits of tk are the same for all k 10.

Answer 1

Answer:

87

Step-by-step explanation:

Consider the sequence t0 = 3; t1 = 3^3; t2 = 3^33 ; t3 = 3^333

; : : : defined by t0 = 3 and

tn+1 = 3^tn for n 0.  

What are the last two digits in t3 = 3333

?  

t0 = 3

t1 = 3³ = 27   (27 = 20*1 + 7)  

t2 = 3²⁷ = 7625597484987  ( 20 * 381279874249 + 7)

t3 = 3⁷⁶²⁵⁵⁹⁷⁴⁸⁴⁹⁸⁷

3²⁰ = 3⁴ˣ⁵  ends with 1

3⁷ = 2187

end with 87

Last two digits of t3 = 3⁷⁶²⁵⁵⁹⁷⁴⁸⁴⁹⁸⁷  = 87