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Transcription:

CHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequece is a fuctio whose domai is the set N of atural umbers. A sequece whose rage is a subset of R is called a real sequece. Geeral A.P. is, a, a + d, a + d,... a = a + ( )d = th term S = Sum of first terms of A.P. a l where l = last term. = a d = If a, b, c are i A.P. the a ± k, b ±k, c ± k are i A.P., ak, bk, ck are also i A.P., k 0 Three umbers i A.P. a d, a, a + d Arithmetic mea betwee a ad b is a b. If A, A, A,...A are iserted betwee a ad b, such that the resultig sequece is A.P. the, b a A a

S k S k = a k a m =, a = m a r = m + r S m = S S m + = 0 S p = q ad S q = p S p + q = p q I a A.P., the sum of the terms equidistat from the begiig ad from the ed is always same, ad equal to the sum of the first ad the last term G.P. (Geometrical Progressio) a, ar, ar,...(geeral G.P.) a = ar a r S, r r Geometric mea betwee a ad b is ab Reciprocals of terms i GP always form a G.P. If G, G, G,...G are umbers iserted betwee a ad b so that the resultig sequece is G.P., the k b G k a, k a I a G.P., the product of the terms equidistat from the begiig ad from the ed is always same ad equal to the product of the first ad the last term. If each term of a G.P. be raised to some power the the resultig terms are also i G.P. Sum of ifiite G.P. is possible if r < ad sum is give by a r r r

r r 6 r r VERY SHORT ANSWER TYPE QUESTIONS ( MARK). If th term of a A.P. is 6 7 the write its 50 th term.. If S = +, the write a. Which term of the sequece,, 0, 7,... is 6? 4. If i a A.P. 7 th term is 9 ad 9 th term is 7, the fid 6 th term. 5. If sum of first terms of a A.P is + 7, write its th term. 6. Which term of the G.P.,,,,,... is? 4 04 7. If i a G.P., a + a 5 = 90 ad if r = fid the first term of the G.P. 8. I G.P.,, 4,..., 8, fid the 4 th term from the ed. 9. If the product of cosecutive terms of G.P. is 7, fid the middle term 0. Fid the sum of first 8 terms of the G.P. 0, 5, 5,.... Fid the value of 5 / 5 /4 5 /8... upto ifiity.. Write the value of 0.. The first term of a G.P. is ad sum to ifiity is 6, fid commo ratio. 4. Write the th 5 7 term of the series,... 7. 8. 9.

5. Fid a 5 of the series whose th term is +. 6. I a ifiite G.P., every term is equal to the sum of all terms that follow it. Fid r 7. I a A.P., 8,, 4,... fid S S SHORT ANSWER TYPE QUESTIONS (4 MARKS) 8. Write the first egative term of the sequece 0, 9,8,7,... 4 4 9. Determie the umber of terms i A.P., 7,,... 407. Also, fid its th term from the ed. 0. How may umbers are there betwee 00 ad 500, which leave remaider 7 whe divided by 9.. Fid the sum of all the atural umbers betwee ad 00 which are either divisible by or by 5.. Fid the sum of the sequece,. If i a A.P. 5 0,,,,, 6 a 5 a fid a 7 a 7 4 0 7 4. I a A.P. sum of first 4 terms is 56 ad the sum of last 4 terms is. If the first term is the fid the umber of terms. 5. Solve : + 6 + + 6 +... + x = 48 6. The ratio of the sum of terms of two A.P.'s is (7 ): ( + ), fid the ratio of their 0 th terms. 7. If the I st, d ad last terms of a A.P are a, b ad c respectively, the fid the sum of all terms of the A.P. 8. b c a c a b a b c If,, are i A.P. the show that a b c,, are also i A.P. [Hit. : Add to each term] a b c

9. If A = + r a + r a +... up to ifiity, the express r i terms of a & A. 0. Isert 5 umbers betwee 7 ad 55, so that resultig series is A.P.. Fid the sum of first terms of the series, 0.7 + 0.77 + 0.777 +.... The sum of first three terms of a G.P. is 5 ad sum of ext three terms is 0. Fid the sum of first terms.. Prove that, 0.0 7 5 [Hit : 0.0 = 0.0 + 0.00 + 0.000 +... Now use ifiite G.P.] LONG ANSWER TYPE QUESTIONS (6 MARKS) 4. Prove that the sum of umbers betwee a ad b such that the resultig a b series becomes A.P. is. 5. A square is draw by joiig the mid poits of the sides of a square. A third square is draw iside the secod square i the same way ad the process is cotiued idefiitely. If the side of the first square is 5 cm, the fid the sum of the areas of all the squares so formed. 6. If a, b, c are i G.P., the prove that a b b c b [Hit : Put b = ar, c = ar ] 7. Fid two positive umbers whose differece is ad whose arithmetic mea exceeds the geometric mea by. 8. If a is A.M. of b ad c ad c, G, G, b are i G.P. the prove that G G abc 9. Fid the sum of the series,..4 + 5.7.8 + 9.. +... upto terms. 40. Evaluate 0 r r

ANSWERS. 9.. 0 th 4. 0 5. 4 + 5 6. th 7. 9 8. 64 9. 0.. 5.. 4. 5. 5 6. 7. + 5 8. 0 8 6 0 r 4 9. 0, 67 0.. 7999. 6. 5 4. 5. 6 6. : 7 7. b c a a c b a 9. A A a

7 0. 5,,, 9, 47. 9 0. 5 7 8 5. 450 cm 7. 6, 4 9. 48 6 4 40. 0

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