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 + a5 = 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 term of the series, 5 7... 7. 8. 9. 5. Fid the umber of terms i the A.P. 7, 0,,...,. 6. I a A.P., 8,, 4,... fid S S 7. Fid the umber of squares that ca be formed o chess board? 8. Fid the sum of give terms:- Page www.lltes.com +9 997 64 906
(a) + + + + 9 + 0. (b) + + +.+ 9 + 40 (c) 8 + 8 + 8 + + 89 + 90 (d) 5 + 5 + 5 +..+ 59 + 60 9. (a) If a, b, c are i A.P. the show that b = a+c. (b) If a, b, c are i G.P. the show that b a.. c 0. If a, b, c are i G.P. the show that i G.P. a b, ab bc, b c are also SHORT ANSWER TYPE QUESTIONS (4 MARKS). Fid the least value of for which + + +... + > 000 5. Fid the sum of the series (+ x) + ( + x + x ) + ( + x + x + x ) +.... Write the first egative term of the sequece 0, 9,8,7,... 4 4 4. Determie the umber of terms i A.P., 7,,... 407. Also, fid its th term from the ed. 5. How may umbers are there betwee 00 ad 500, which leave remaider 7 whe divided by 9. 6. Fid the sum of all the atural umbers betwee ad 00 which are either divisible by or by 5. 7. Fid the sum of the sequece, 5 0,,,,, 6 8. If i a A.P a 5 a fid a 7 a 7 4 0 7. Page www.lltes.com +9 997 64 906
9. 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. 0. Solve: + 6 + + 6 +... + x = 48. The ratio of the sum of terms of two A.P.'s is (7 ): ( + ), fid the ratio of their 0 th terms.. 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. b c a c a a a b c. If are i A.P. the show that,, a b c a b c are also i A.P. [Hit. : Add to each term] abc 4. The product of first three terms of a G.P. is 000. If 6 is added to its secod term ad 7 is added to its third term, the terms become i A.P. Fid the G.P. 5. If the cotiued product of three umbers i G.P. is 6 ad the sum of their products i pairs is 56, fid the umbers. 6. Fid the sum to ifiity of the series: 5 7... 7. If A = + r a + r a +... up to ifiity, the express r i terms of a & A. 8. Fid the sum of first terms of the series 0.7 + 0.7 + 0.777 +... a a b b 9. If x a... ; y b... ad r r r r c c z c... 4 r r prove that xy ab. z c Page 4 www.lltes.com +9 997 64 906
40. 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. 4. Prove that 7 0.00 5 [Hit: 0.0 = 0.0 + 0.00 + 0.000 +... Now use ifiite G.P.] 4. If log, log( ) ad log ( +) are i A.P. Show that 4. If a, b, c are i G.P. that the followig are also i G.P. (i) (ii) a, b, c a, b, c (iii) a, b, c are i G.P. 44. If a, b, c are i A.P. that the followig are also i A.P: (i) (ii) (iii),, bc ca ab b + c, c + a, a + b 45. If the umbers,, a b c b c a c a b,, b c c a a b 46. Show that: are i A.P. log 5 log a, b, c are give to be i A.P., show that are i A.P. 5 0.56 990 Page 5 www.lltes.com +9 997 64 906
LONG ANSWER TYPE QUESTIONS (6 MARKS) 47. Prove that the sum of umbers betwee a ad b such that the ( a b) resultig series becomes A.P. is 48. 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. 49. If a, b, c are i G.P., the prove that a b b c b [Hit : Put b = ar, c = ar ] 50. Fid two positive umbers whose differece is ad whose arithmetic mea exceeds the geometric mea by. 5. If a is A.M. of b ad c ad c, G, G, b are i G.P. the prove that G G abc 5. Fid the sum of the series,..4 + 5.7.8 + 9.. +... upto terms. 5. Evaluate 0 r r. 54. The sum of a ifiite G.P. is 57 ad the sum of the cubes of its term is 9747, fid the G.P. Page 6 www.lltes.com +9 997 64 906
9 8 7 9 9 0 () (0) () (0)... 0() k.(0). the fid 55. If the value of k such that k N. 56. Fid the sum of first terms of the series 7 5... 4 8 6 terms. 57. Three positive umbers form a icreasig G.P. If the middle term i the G.P. is doubled, the ew umbers are i A.P. the fid the commo ratio of the G.P. 58. Show that if the positive umber a, b, c are i A.P. so are the umbers,, are i A.P. a c c a a b 5 7 9 59. Fid the sum of the series:-.... 4 8 6 a, a, a,... a a 0i N are i A.P. where 60. If i. the show that... a a a a a a a a a a 4 6. If the sum of first terms of a A.P. is c. the prove that the c 4 sum of squares of these terms is. 6. Let p ad q be the roots of the equatio x x A 0 ad let r ad s be the roots of the equatio x 8x B 0 if p< q< r< s are i A.P. the prove that A= ad B= 77. 6. If S, S, S... S are the sums of ifiite geometric series whose first terms are,,, ad whose commo ratios are,,... 4 respectively, the show that the value of S S S... S 4 6 Page 7 www.lltes.com +9 997 64 906
4 A.... 4 4 4 4 4 ad B Fid at least odd atural umber o, so that B A o. 64. Let th th 65. If p, q ad r th A. terms of a A.P. ad G.P. are equal ad are x, y yz zx x y ad z respectively, the prove that x. y. z ANSWERS. 9.. 0 th 4. 0 5. 4 + 5 6. th 7. 9 8. 64 9. 0. 0 8. 5.. 4. ( 6)( 0) 5. 9 6. + 5 7. 04 8. (a) 55 (b) 55 (c) 855 (d) 555 Page 8 www.lltes.com +9 997 64 906
x ( x ). = 7. x ( x). 4. 0, 67 4 5. 6. 7999 7. 8. 9. 0. 6. :7. ( b c a)( a c) ( b a) 4. 5, 0, 0,...; or 0, 0, 5, 5. 8, 6, ; or, 6, 8 6. 6 7. 8. 7 9 0 8 A A a 40. 5 7 48. 450 cm² 50. 6, 4 5. ( ) (48 6 4) 5. 0 54. 8 76 9,,,... 9 55 k = 00 56. 57. r 59. 64. 0 7 9 Page 9 www.lltes.com +9 997 64 906