Progression 1) Arithmetic Progression 2) Geometric Progression 3) Harmonic Progression Arithmetic progression: Arithmetic Progression is a Progression of the form a, a + d, a + 2d, a + 3 d, a + 4d…….. First term =a Common difference=d Nth term

$$a_n$$ = a + ( n -1 )d a $$\rightarrow$$ first term d $$\rightarrow$$ common difference $$S_n = \frac{n}{2}$$ [ 2a + ( n - 1 ) d ] $$S_n= \frac{n}{2}$$ [ a + l ] L $$\rightarrow$$ last term 1 = a + ( n – 1 )d (i) The taxi fare after each km. when the fare is RS. 15 for the first km. and Rs 8 additional for additional km Solution: a = 15 km d = 8 km a+d= 15+8 =23 a+2d= 15+ 8(2)=31 a+3d= 15+ 8 (3) =39 15, 23,31,39,…….   (i)  a= 4 ; d=-3 Find the first 5 terms of the progression Solution: a=4 2^nd term = a+d = 4-3=1 3^rd term= a+2d = 4-6=-2 4^th term= a+ 3d =4-9=-5 5^th term =a+4d=4-12=-8 First five terms of progression 4, 1, -2,-5, -8 (i) for the following A.P find the first term and the common difference 3,1,-1,-3,………. Solution: a=3 d= $$a_2 - a_1$$ =1-3=-2 (ii) $$\frac{1}{3}, \frac{5}{3}, \frac{9}{3}$$ ,………. Solution: a= $$\frac{1}{3}$$ d= $$a_2-a_1$$ = $$\frac{5}{3}-\frac{1}{3}$$ = $$\frac{4}{3}$$ Which of the following are AP’s ? If they form and A.P find the common difference and 3 more terms (i) 2,4,8,16,……. a=2 d = $$a_2 –a_1$$ =4-2=2 d= $$a_3-a_2$$ =8-4=4 $$d_1 =2 d_2 =4$$ 2,4,8,16,…………….. Do not form an arithmetic progression (ii) 2, $$\frac{5}{2} ,3, \frac{7}{2}$$ ……….. a=2 d= $$a_2-a_1= \frac{5}{2} -2 = \frac{5-4}{2}=\frac{1}{2}$$ d= $$a_3 - a_2 = 3- \frac{5}{2}= \frac{6-5}{2}=\frac{1}{2}$$ d= $$\frac{1}{2}$$ This forms an arithmetic progression $$\frac{7}{2}+ \frac{1}{2}=\frac{8}{2}=4$$ 4+ $$\frac{1}{2}= \frac{9}{2}$$ $$\frac{9}{2}+ \frac{1}{2}= \frac{10}{2}$$ = 5 The next three terms are 4, $$\frac{9}{2}$$ ,5 Fill in the  following table. Given that a is the first term the common difference d, an is the last term

	a	d	n	$$a_n$$
I	7	3	8	….
II	-18	…..	10	0
III	…..	-3	18	-5
IV	-18.9	2.5	……	3.6
V	3.5	0	105

(I)a=7 d=3 n=8

$$a_n$$ = a+ (n-1)d =7+(8-1)3 =7+21 =28 (II) a=-18 d=? n=10 $$a_n$$ =0 $$a_n$$ = a+ (n-1)d 0=-18+(10-1)d 0=-18 +9d 9d=18 d=18/9=2 (III) a=? d=-3 n=18 $$a_n$$ =-5 -5 =a+ (18-1)-3 -5=a+(17)-3 -5=a-51 a=51-5 =46   (IV) a=-18.9  d=2.5 n=? , $$a_n$$ =3.6 3.6=-18.9+(n-1)2.5 3.6=-18.9+2.5n-2.5 2.5n=3.6+18.9+2.5 2.5n=25 n= $$\frac{25}{2.5}$$ =10   (V) a=3.5 d=0 n=105 $$a_n$$ =3.5 + (105-1)0 $$a_n$$ =3.5 30^th term of A.P 10,7,4,…….. a=10 d= $$a_2-a_1$$ =7-10=-3 n=30 $$a_n$$ = a+ (n-1)d =10+(30-1)-3 =10+(29)-3 =10-87=-77 30^th term of the A.P =-77 18,13,8,3 a=a $$a_2$$ =13 =a+d $$a_3$$ =a+2d=x $$a_4$$ =a+3d=3 a+3d=3 a+d=13

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2d=-10 d=-5 a+d=13 a-5=13 a=18 a=18 a+2d=18+2 (-5) =8 -4,-2,0,2,4,6 First term=-4 Sixth term

$$a_6$$ =6 Common difference=d $$a_n$$ =a+(n-1)d a=-4 n=6 $$a_n$$ =6 6=-4+(6-1)d 10=5d d=10/5=2 2^nd term=-4+2=-2 3^rd term =-4+4=0 4^th term =-4+6=2 5^th term =-4+8=4 Which term of the A.P 3,8,13,18,…… is 78 a=3 d= $$a_2-a_1$$ =8-3=5 $$a_n$$ =78 78=3+(n-1)5 78=3+5n-5 5n=78+2=80 n= $$\frac{80}{5}$$ = 16 16^th term of the A.P is 78 Find the number of term in the following A.P 7,13,19,……,205 a=7 d= $$a_2-a_1$$ =13-7=6 $$a_n$$ =205 205=7+(n-1)6 205=7+6n-6 205=6n+1 6n=204 n= $$\frac{204}{6}$$ =34 no.of terms in the A.P =34    

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