Serena used 144 as the first term in a pattern. To get terms after the first, she used the rule "divide by 2, divide by 3, divide by 4, etc." 144, 72, 24, 6, ? If Serena continued her pattern according to the same rule, which term should she have written next? Thx!

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .