After Robert Fulton invented the steamboat, he tried it out on the Hudson River.
Answer:
Modal of permission: Students may be allowed to use their personal computers in class to take notes and read digital books.
Modal of obligation: Students and teachers have to recycle papers.
Modal of prohibition: The school does not have to leave the lights on when no one is in the classroom
Explanation:
Modals of permission are used in a sentence to inform or ask if an action is allowed. These modals are can, may, and could. May and could are more formal than can.
Modals of obligation are used in a sentence to inform of something compulsory. Must is a modal of obligation use for a personal obligation like I must study for the exam, or rules like you must wear gloves in the laboratory. Have to, is also a modal of obligation, but it expresses general obligation like Students have to study hard for the exam.
Modals of prohibition are in sentences that express something that is not allowed. They are can not and must not. For example, you can not smoke inside this building.
It is clear that a(n)=2^(1-2^(-n)). In fact, for n=1 this produces 2^(1-1/2)=sqrt(2)=a1 and if it is true for a(n) then a(n+1) = sqrt (2 * 2^(1-2^(-n))) = sqrt(2^(2-2^(-n))) = 2^(1-2^(-(n+1))) (a) clearly 2^(1-2^(-n))<2<3 so the sequence is bounded by 3. Also a(n+1)/a(n) = 2^(1-2^(-n-1) - 1+2^(-n)) = 2^(1/2^n - 1/2^(n+1)) = 2^(1/2^(n+1)) >1 so the sequence is monotonically increasing. As it is monotonically increasing and has an upper bound it means it has a limin when n-> oo (b) 1-1/2^n -> 1 as n->oo so 2^(1-2^(-n)) -> 2 as n->oo
