Exercise 4.1 2. Establish each of the following for all n ≥ 1 by the Principle of Mathematical Induction. Solution a) S(n): ==‚ S(1): = = =1‚ So S(1) is true. Assume S(k): = Consider S(k+1) = = +=-1+= -1. Hence‚ it follows that S(k)⇒S(k + 1) is true for all n ∈ Z+ by the Principle of Mathematical Induction. b) S( n) for n=1‚ = 2 = 2+(1-1). So S(1) is true. Inductive Step: assume S(k)is true‚ for some (particular) k ∈ Z+—that is‚ assume that =2+(k-1). For n=k+1‚ = + (k+1)
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velocity‚ and acceleration are vector quantities‚ while speed (the magnitude of velocity)‚ time‚ and mass are scalars. To qualify as a vector‚ a quantity having magnitude and direction must also obey certain rules of combination. One of these is vector addition‚ written symbolically as A + B = C (vectors are conventionally written as boldface letters). Geometrically‚ the vector sum can be visualized by placing the tail of vector B at the head of vector A and drawing vector C—starting from the tail of A
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done first. When you have more than one set of brackets‚ do the inner brackets first ie starting our work with the innermost pair‚ moving outward. 3. Exponents (or radicals) must be done next. 4. Multiplication and division are done next . 5. Addition and subtraction must be done next. The acronym which can be used to remember this order is Big Elephants Destroy Mice And Snails (Brackets‚ Exponents‚ Divide‚ Multiply‚ Add‚ Subtract) If the order of operation is not followed‚ a lot of problem
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following connection words are listed by function. To add details Accordingly‚ additionally‚ again‚ also‚ and‚ as‚ as well as‚ aside from‚ at any rate‚ besides‚ equally important‚ especially‚ for example‚ for instance‚ further‚ furthermore‚ in addition‚ in other instances‚ in particular‚ in some cases‚ indeed‚ moreover‚ to illustrate‚ that‚ which‚ who To show cause and effect As a consequence of‚ as a result of‚ for this reason‚ because‚ consequently‚ for‚ hence‚ in spite of‚ on account of
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with minuends up to 900 without regrouping. References: BEC 3.1 p. 11 TM – Mathematics 2000 pp. 220-222; TX Mathematics 2000 pp.29-300 Materials: flashcards ‚ textbook III.Learning Activities A. Activities: 1. Drill : addition without regrouping. 2. Review: subtracting with regrouping. 3. Checking of assignment. B. Lesson proper: Motivation : 1. Present on a chart on this word problem. A park needs to be planted with 124 trees. The boys scouts planted
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was the first mechanical adding machine. It was 36cm long; 13cm wide; and 8cm wide. It had eight movable parts that added up to eight‚ figured long sums‚ and used base 10. Each drum had two sets of rows. The black row of numbers was dealing with addition‚ and the red row of numbers was dealing with subtraction. The machine was able to convert and exchange rates for different currencies. The pascalines had metal wheel dials that turned to the appropriate numbers‚ answers appeared in the boxes in the
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with the date‚ time‚ and also a slide to unlock slider. Those few improvements to the overall OS were nice‚ but the slide to unlock from the original OS 6 was still there and so were the large digital clock at the top. Another difference is the addition of Control Center. Control Center is a slider at the bottom of the screen that you slide up to use specific features of IOS 7‚ that in the older OS 6 you would have had to go to the settings menu. The features of Control Center include a slider for
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medicine for everything. But they make a mistake‚ as no sum of cash can prevent us from old age‚ loneliness‚ diseases‚ accidents‚ death or stuff like this. And it’s useful to remember that money is a fragile thing‚ which we can lose at any time. In addition‚ wealth makes people less sensitive. They aren’t able to notice both poverty‚ which is around them‚ and needs of others. However‚ money is the basic means‚ which lets people make their dreams come true - the dreams‚ which have been unavailable since
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simplified versions have been seen with one bead above and four to five below the horizontal bar. Even in today’s world of technology the abacus is still used in many cultures such as China. Tests have shown that performing math problems such as addition and subtraction on an abacus are faster than an electronic calculator. Today’s college student is use to computers and calculators that automatically give us an answer‚ provided that we put in the right information. While working with an Abacus
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practical activities and discussion begin to use the vocabulary involved in adding and subtracting * Use language such as ‘more’ or ‘less’ to compare two numbers * Find one more or one less than a number from one to 10 * Begin to relate addition to combining two groups of objects and subtraction to ‘taking away. * Shape‚ space and measures Expectations: * Use language such as ‘greater‚ ‘smaller’‚ ‘heavier’ or ‘lighter’ to compare quantities * Talk about‚ recognise and recreate
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