maths.pdf

Number Patterns 1 cm 2 cm 3 cm 4 cm See the squares. What are their perimeters? And areas? As the lengths of the sides go 1 cm, 2 cm, 3 cm, 4 cm,... the perimeters are 4 cm, 8 cm, 12 cm, 16 cm,... And the areas 1 sq.cm, 4 sq.cm, 9...

Number Patterns 1 cm 2 cm 3 cm 4 cm See the squares. What are their perimeters? And areas? As the lengths of the sides go 1 cm, 2 cm, 3 cm, 4 cm,... the perimeters are 4 cm, 8 cm, 12 cm, 16 cm,... And the areas 1 sq.cm, 4 sq.cm, 9 sq.cm, 16 sq.cm,... Let’s look at the numbers alone. The lengths of the sides are just the natural numbers, written in order: 1, 2, 3, 4,... The perimeters form the multiples of 4, in order: 4, 8, 12, 16,... And the areas form the perfect squares in order: 1, 4, 9, 16,... Mathematics X What about their diagonals? Write those numbers also. How about increasing the lengths of sides in steps of half a centimetre instead of one centimetre? 1 1 1 cm 1 cm 2 cm 2 cm 2 2 1 1 Side 1, 1 , 2, 2 ,... 2 2 Perimeter 4, 6, 8, 10,... 1 1 Area 1, 24, 4, 64,... 3 5 Diagonal 2, 2 2, 2 2, 2 2,... A set of numbers written like this, as the first, second, third and so on, according to a particular rule is called a    . We can make another sequence with squares. Imagine a square of side 1 metre. Joining the midpoints of the sides, we get another square: 1 metre What is the area of this smaller square? Its diagonal is 1 metre; and we know that the area of a square is half the square of its diagonal (The section, Rhombus of the lesson Areas of Quadrilaterals, in the Class 8 textbook). So, the area of the small square is half a square metre. 8  Arithmetic Sequences Continuing, the area is halved each time: 1 metre 1 metre 1 metre What number sequence do we get from this? 1 1 1 1, , , , 2 4 8... We get sequences from physics also. The speed of an object falling from a height increases every instant. If the speed at  seconds is taken as  metres per second, the time–speed equation is 9.8 If the distance travelled in  seconds is taken as  metres, then the time– distance equation is Different kinds of sequences = 4.9 The word sequence in ordinary So, we get two sequences from this: language means things occurring one after another, in a definite order. In Time 1 , 2, 3, 4,... mathematics, we use this word to Speed 98. , 19 6. , 29 4. , 39 2,.... denote mathematical objects placed as the first, second, third and so on. Distance 49. , 19 6. , 44 1. , 78 4,.... The objects thus ordered need not always be numbers. Let’s have another example from physics. The density of For example, we can have a iron is 7.8 gm/cc. This means an iron cube of volume 1 cubic sequence of polygons as given below: centimetre would weigh 7.8 grams. So, iron cubes of sides 1 centimetre, 2 centimetres, 3 centimetres and so on, would have volumes 1 cubic centimetre, 8 cubic centimetres, 27 cubic centimetres and so on; and should weigh 7.8 grams, Or a sequence of polynomials as 62.4 grams, 210.6 grams and so on. Writing these as number 1 + x, 1 + x2. 1 + x3,... sequences, we get The arrangements of words of a language in alphabetical order is also a sequence. 9 Mathematics X Side 1, 2 , 3 ,... Volume 1, 8 , 27,... Weight 78. , 62.4 , 210.6 ,... We can form sequences from peculiarities of pure numbers, instead of numbers as measures. For example, the prime numbers written in order gives the sequence 2 3 5 7 11 13, , , , , ,... 21 The digits in the decimal form of 37 , written in order is the sequence 5 6 7 5 6 7 5 6 7,... , , , , , , , , If we take π instead, we get this sequence: 3 1 4 1 5 9 2 6,... , , , , , , , The same sequence can be described in different ways. For example, this is the sequence of natural numbers ending in 1: 1 11 21 31 , , , ,... We can also say that this is the sequence of natural numbers which leave remainder 1 on division by 10. (1 ) Make the following number sequences, from the sequence of equilateral triangles, squares, regular pentagons and so on, of regular polygons: Number of sides 3, 4, 5,... Sum of inner angles Sum of outer angles One inner angle One outer angle (2) Look at these triangles made with dots. How many dots are there in each? Compute the number of dots needed to make the next three triangles. 10  Arithmetic Sequences (3) Write down the sequence of natural numbers leaving remainder 1 on division by 3 and the sequence of natural numbers leaving remainder 2 on division by 3. (4) Write down the sequence of natural numbers ending in 1 or 6 and describe it in two other ways. (5) A tank contains 1000 litres of water and it flows out at the rate of 5 litres per second. How much water is there in the tank after each second? Write their numbers as a sequence. Algebra of sequences We have seen that the perimeters of squares of sides 1 centimetre, 2 centimetres, 3 centimetres and so on, form the sequence 4 8 12 , , ,... The numbers forming a sequence are called its . Thus 4, 8, 12,... are the terms of the above sequence. More precisely, 4 is the first term, 8 is the second term, "        12 is the third term and so on.       GeoGebra We can write this as given below: #$  %&  Position 1, 2, 3 ,... '  () (%& * +,-*    &    Term 4, 8, 12 ,... $  +,-   What is the 5 term? The 20 term?        %&    What is the relation between the positions and the terms?     $     '        Each term is four times its position.   Using a bit of algebra, we can put it like this: '  () (%& ./* ,* The  term of the sequence is 4           +01       Usually the terms in a sequence are written in algebra as ,   ,/+  ,  ,... or , , ,...    .//   So, we can shorten the above rule further:      2"/ .,  2 n = 4 11