A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mn be the mass of the radioactive material at the end of the nth decade, where the initial mass of the material is Mo = 40 g. (a) Write out the first five terms of the sequence. (b) Find an explicit formula for the terms of the sequence. (c) Estimate the limit of the sequence or state that it does not exist.

Please answer no. 4 only

A savings account with an interest rate r, which is compounded n times per year, and begins with P as the principal (initial amount), has the discrete nt compounding formula A (t) = P(1+)". This is n because we multiply the amount by itself plus a small amount, determined by the interest rate, and the account grows each time the compounding occurs. For continuous compounding, we use the formula A (t) = Pert , and if we have seen this formula before, we may not have gotten a satisfactory answer as to why we use it, other than some vague notion of "compounding infinity times per year". In this exercise, we'll use Bernoulli's Rule to find the connection. It might be helpful to review the "Indeterminate Powers" section of the video before beginning. Why can we write nt lim,→00 P(1+ )"t P limn¬∞ (1+)™ ? n

A client’s orders are growing consistently at x% per year.     A) If three years from now orders are 50% higher than today, what is the annual growth rate (x)?     B) Write an expression (in terms of x) for how many years it will take until orders are double what they are currently

Let a, = 2" cos(n7) for n21. Write out the first 8 terms of the sequence (a, through ag). Determine lim a, . Note: You must show all your work and justify your steps carefully. а. b. Determine if Ea, converges (absolutely or conditionally) or diverges. Note: You must show с. all your work and justify your steps carefully.