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lukranit [14]
3 years ago
14

A sequence{an} is given by a1=sqrt(2), an+1=sqrt(2+an).

English
1 answer:
MrMuchimi3 years ago
3 0
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
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Introduction:

Everybody loves a clean environment but very few have the ability to maintain cleanliness. The ability to ensure cleanliness and perform cleanliness is tasking and requires a lot of work. Due to the importance of cleanliness, most people struggle to keep their environments clean at all times. It can be a habitual process as it require to be done on a daily basis. Personal hygiene for example is an essential aspect of an individual’s life because it affects social relations, self-perception and health in general. It is therefore important to practice cleanliness at all times.

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In the workplace, the same concept applies because productivity is influenced by the environment. A clean environment will enhance productivity because people will work with a positive mindset that is free of stress. Also, achievements will be more pronounced in a clean environment compared to a dirty environment. In health, cleanliness is important because hygienic practices contribute to good health of individuals. A dirty environment predisposes people to diseases, which affects their ability to perform activities and it results in less productivity.

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e sick. As a result of this, we should do our best to always try to take good care of our health through our cleanliness every day. Some of the habits we should endeavour to imbibe include us always washing our hands very well and with soap anytime we eat or before we eat anything, we should also try to always make sure our body and most especially our face is clean by bathing frequency, we should also ensure that we take proper care of all our clothes and try to wear only clothes that are properly and well washed and clean; all of these should be done to keep a very good mental and physical health. Cleanliness also does a great deal to help boost our self-respect and level of confidence and also earn respect from other people. Cleanliness is a very good habit that can always make us happy and proud of ourselves in the society.

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