Welcome to� X Mathematics class�T.V.N.RAO TGT(Maths) �J.N.V.Khammam (T.S)�Topic : ARITHMETIC � PROGRESSIONS�Sub-Topic : Introduction to A.P�

ARITHMETIC PROGRESSIONS

• 2 files are sent on the above topic

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1. Power Point presentation with 25 slides + audio

2. Topic presentation by 3 Videos duration of 10-15 minutes each.

Proper care has been taken in making this project. However if any corrections please bring to my knowledge.

Let us watch the following patterns�a) 2, 4, 1, 5, 7, -3, 1, 5……..��b) 2, 4, 6, 8, 10, . . . . …..��c) 1, 2, 4, 8, 16,…………..�� d) 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5,……�� e) -3, -2, -1, 0, 1, 2, 3,……….�� f)) 1, 1 ½ , 2, 2 ½ , 3, 3 ½ , . . . . . ..

Let us watch coloured patterns�2, 4, 1, 5, 7, -3, 1, 5………��2, 4, 6, 8, 10 , . . . . …..��1, 2, 4, 8, 16,………….

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1, 3, 9, 27 ,………..��-3, -2, -1, 0, 1, 2, 3,……….��1, 1 ½ , 2, 2 ½ , 3, 3 ½ , . . . . . ..

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They follow certain rules, systematically arranged

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Le us now watch these patterns�1,2,4,8,16,………….��1, 3, 9, 27,………..��

Let us watch patterns��2, 4, 6, 8, 10, . . . . …..��-3, -2, -1, 0, 1, 2, 3,……….��1, 1 ½ , 2, 2 ½ , 3, 3 ½ , . . . . . ..�Every successor is obtained by adding a fixed number to its predecessor.���-

The pattern or Sequence is called PROGRESSION�Each element/number in this pattern is a TERM�This kind of arrangement of numbers is defined as�ARITHMETIC PROGRESSION�Difference of any two successive terms is constant�let us see example�2, 4, 6, 8,….. �

�2, 4, 6, 8,….. � 4-2=2� 6-4=2� 8-6=2�2, 4, 6, 8 , …… are said to be in ARITHMETIC PROGRESSION(A.P.)

Let us see example�2, 4, 6, 8,….. � 4-2=2� 6-4=2� 8-6=2

Here, difference 2 is fixed

This fixed difference is called COMMON DIFFERENCE(d)

Now let us generalize A.P�a, a+d, a+2d, a+3d,…………�the difference of any two successive terms is d�So, a, a+d, a+2d, a+3d,…… are in A.P.�here, first term = a�Common difference = d

a, a+d, a+2d, a+3d,…… are in A.P.�here, first term = a�Common difference = d�let, t₁, t₂ , t₃ , t₄ ,.…..are terms�First term = t₁ = a�second term = t₂ = a + d�third term = t₃ = a + 2d

Let t₁, t₂ , t₃ , t₄ ,.….. A.P�First term = t₁ = a�second term = t₂ = a + d�third term = t₃ = a + 2d�…………………………………………�…………………………………………�Tenth term = t₁₀ = a + 9d �now we can generalize the nth term as t_n�t_n = a+ (n-1) d

t_n = a+ (n-1) d�let us find 10^th term in �5, 10, 15, ……….�Here, first term = a = 5�Common difference = d = 5� t_n = a+ (n-1) d�t₁₀ = 5+ (10-1) 5 = 5 + 9(5)� = 5 + 45 = 50

t_n = a+ (n-1) d�let us find 27^th term in �5, 2, -1, -4, -7, ……….�Here, first term = a = 5�Common difference = d = 2-5 = -3� t_n = a+ (n-1) d�t₂₇ = 5+ (27-1) (-3) = 5 + (26)(-3)� = 5 - 78 = -73

t_n = a+ (n-1) d�let us find n^th term in �5, 2, -1, -4, -7, ……….�Here, first term = a = 5�Common difference = d = 2-5 = -3� t_n = a+ (n-1) d�t_n = 5+ (n-1) (-3) = 5 – 3n + 3� = 8 - 3n �

Find the sum of 51 terms whose second and third terms are 14 and 18 respectively�given second term = t₂ = 14� eighteenth term = t₃ = 18�t₂ = 14 ; a + d = 14�t₃ = 18; a + 2d = 18�� a + d = 14 ………… (i)� a + 2d = 18 …… (ii)� �

Thank for watching� X Mathematics class�T.V.N.RAO TGT(Maths) �J.N.V.Khammam (T.S)

My special thanks to all officials��