Page 1 1) Find 6 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 6 6 6 1 ? ? ? ? a 216 ) 1 ( 5 6 ? ? ? a = -216 2) Find 13 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 13 13 13 1 ? ? ? ? a 2197 ) 1 ( 12 13 ? ? ? a = 2197 3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term. Solution: Here, common difference (d) = -10 First term 200 1 ? a d n a a n ) 1 ( 1 ? ? ? ) 10 )( 1 20 ( 200 20 ? ? ? ? a 10 20 ? a 4) An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term Solution: We use nth term formula d n a a n ) 1 ( 1 ? ? ? for the 6 th term ) 1 6 ( 10 52 1 6 6 ? ? ? ? a a a 50 52 1 ? ?a Therefore, 2 50 52 1 ? ? ? a Now, we know the first term and the common difference, we use the nth term formula to find the 15 th term. 142 ) 1 15 ( 10 2 15 ? ? ? ? a Arithmetic Progressions Page 2 1) Find 6 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 6 6 6 1 ? ? ? ? a 216 ) 1 ( 5 6 ? ? ? a = -216 2) Find 13 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 13 13 13 1 ? ? ? ? a 2197 ) 1 ( 12 13 ? ? ? a = 2197 3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term. Solution: Here, common difference (d) = -10 First term 200 1 ? a d n a a n ) 1 ( 1 ? ? ? ) 10 )( 1 20 ( 200 20 ? ? ? ? a 10 20 ? a 4) An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term Solution: We use nth term formula d n a a n ) 1 ( 1 ? ? ? for the 6 th term ) 1 6 ( 10 52 1 6 6 ? ? ? ? a a a 50 52 1 ? ?a Therefore, 2 50 52 1 ? ? ? a Now, we know the first term and the common difference, we use the nth term formula to find the 15 th term. 142 ) 1 15 ( 10 2 15 ? ? ? ? a Arithmetic Progressions 5) If x times the x th term of an AP is equal to y times its y th term , Prove that the (x + y) th term of the AP is zero Solution: Let â€˜aâ€™ be the first term andâ€˜dâ€™ be the common difference of an AP By the given condition, y x yt xt ? ? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ........ 0 ) 1 ( 0 ) 1 ( ) ( 0 ) ( ) )( ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y x t x d y x a d y x a y x d y x y x y x a y x d y x y x a y x d y y d x x ya xa d y y ya d x x xa ? 6) Which term of AP 7, 14, 21,â€¦â€¦..is 133 Solution: Here, a = 7 , d = 14 â€“ 7= 7 Let 133 be the th n term of the AP ( Where N n ? ) 133 ? n t 133 ) 1 ( ? ? ? ? d n a ? ? N n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 19 . .......... 19 7 133 133 7 133 7 7 7 133 7 ) 1 ( 7 Therefore, 133 is the 19 th term of the given AP. Arithmetic Progressions Page 3 1) Find 6 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 6 6 6 1 ? ? ? ? a 216 ) 1 ( 5 6 ? ? ? a = -216 2) Find 13 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 13 13 13 1 ? ? ? ? a 2197 ) 1 ( 12 13 ? ? ? a = 2197 3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term. Solution: Here, common difference (d) = -10 First term 200 1 ? a d n a a n ) 1 ( 1 ? ? ? ) 10 )( 1 20 ( 200 20 ? ? ? ? a 10 20 ? a 4) An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term Solution: We use nth term formula d n a a n ) 1 ( 1 ? ? ? for the 6 th term ) 1 6 ( 10 52 1 6 6 ? ? ? ? a a a 50 52 1 ? ?a Therefore, 2 50 52 1 ? ? ? a Now, we know the first term and the common difference, we use the nth term formula to find the 15 th term. 142 ) 1 15 ( 10 2 15 ? ? ? ? a Arithmetic Progressions 5) If x times the x th term of an AP is equal to y times its y th term , Prove that the (x + y) th term of the AP is zero Solution: Let â€˜aâ€™ be the first term andâ€˜dâ€™ be the common difference of an AP By the given condition, y x yt xt ? ? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ........ 0 ) 1 ( 0 ) 1 ( ) ( 0 ) ( ) )( ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y x t x d y x a d y x a y x d y x y x y x a y x d y x y x a y x d y y d x x ya xa d y y ya d x x xa ? 6) Which term of AP 7, 14, 21,â€¦â€¦..is 133 Solution: Here, a = 7 , d = 14 â€“ 7= 7 Let 133 be the th n term of the AP ( Where N n ? ) 133 ? n t 133 ) 1 ( ? ? ? ? d n a ? ? N n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 19 . .......... 19 7 133 133 7 133 7 7 7 133 7 ) 1 ( 7 Therefore, 133 is the 19 th term of the given AP. Arithmetic Progressions 7) Which term of AP 10, 17, 24,â€¦â€¦..is 150 Solution: Here, a = 10 , d =17 â€“ 10 = 7 Let 150 be the th n term of the AP ( Where N n ? ) 150 ? n t n t d n a ? ? ? ? ) 1 ( ? ? N n n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 21 . .......... 21 7 147 147 7 150 7 3 150 7 7 10 150 7 ) 1 ( 10 Therefore, 150 is the 21 th term of the given AP. 8) An arithmetic sequence has its 5 th term equal to 22 and its 15 th term equal to 62. Find its 100 th term Solution: We use the formula for 5 th term and 15 th term ) 1 ......( .......... 4 22 ) 1 5 ( 1 1 5 d a d a a ? ? ? ? ? ) 2 ......( .......... 14 62 ) 1 15 ( 1 1 15 d a d a a ? ? ? ? ? Solving equation (1) and (2) we get value of d Therefore d = 4 Now, plug in the value of d in equation (1) we get the value of 1 a 6 16 22 16 22 1 1 1 ? ? ? ? ? ? ? a a a Now, we use nth term formula to find the 100 th term 402 4 ) 1 100 ( 6 100 100 ? ? ? ? a a Arithmetic Progressions Page 4 1) Find 6 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 6 6 6 1 ? ? ? ? a 216 ) 1 ( 5 6 ? ? ? a = -216 2) Find 13 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 13 13 13 1 ? ? ? ? a 2197 ) 1 ( 12 13 ? ? ? a = 2197 3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term. Solution: Here, common difference (d) = -10 First term 200 1 ? a d n a a n ) 1 ( 1 ? ? ? ) 10 )( 1 20 ( 200 20 ? ? ? ? a 10 20 ? a 4) An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term Solution: We use nth term formula d n a a n ) 1 ( 1 ? ? ? for the 6 th term ) 1 6 ( 10 52 1 6 6 ? ? ? ? a a a 50 52 1 ? ?a Therefore, 2 50 52 1 ? ? ? a Now, we know the first term and the common difference, we use the nth term formula to find the 15 th term. 142 ) 1 15 ( 10 2 15 ? ? ? ? a Arithmetic Progressions 5) If x times the x th term of an AP is equal to y times its y th term , Prove that the (x + y) th term of the AP is zero Solution: Let â€˜aâ€™ be the first term andâ€˜dâ€™ be the common difference of an AP By the given condition, y x yt xt ? ? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ........ 0 ) 1 ( 0 ) 1 ( ) ( 0 ) ( ) )( ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y x t x d y x a d y x a y x d y x y x y x a y x d y x y x a y x d y y d x x ya xa d y y ya d x x xa ? 6) Which term of AP 7, 14, 21,â€¦â€¦..is 133 Solution: Here, a = 7 , d = 14 â€“ 7= 7 Let 133 be the th n term of the AP ( Where N n ? ) 133 ? n t 133 ) 1 ( ? ? ? ? d n a ? ? N n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 19 . .......... 19 7 133 133 7 133 7 7 7 133 7 ) 1 ( 7 Therefore, 133 is the 19 th term of the given AP. Arithmetic Progressions 7) Which term of AP 10, 17, 24,â€¦â€¦..is 150 Solution: Here, a = 10 , d =17 â€“ 10 = 7 Let 150 be the th n term of the AP ( Where N n ? ) 150 ? n t n t d n a ? ? ? ? ) 1 ( ? ? N n n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 21 . .......... 21 7 147 147 7 150 7 3 150 7 7 10 150 7 ) 1 ( 10 Therefore, 150 is the 21 th term of the given AP. 8) An arithmetic sequence has its 5 th term equal to 22 and its 15 th term equal to 62. Find its 100 th term Solution: We use the formula for 5 th term and 15 th term ) 1 ......( .......... 4 22 ) 1 5 ( 1 1 5 d a d a a ? ? ? ? ? ) 2 ......( .......... 14 62 ) 1 15 ( 1 1 15 d a d a a ? ? ? ? ? Solving equation (1) and (2) we get value of d Therefore d = 4 Now, plug in the value of d in equation (1) we get the value of 1 a 6 16 22 16 22 1 1 1 ? ? ? ? ? ? ? a a a Now, we use nth term formula to find the 100 th term 402 4 ) 1 100 ( 6 100 100 ? ? ? ? a a Arithmetic Progressions 9) Determine p, so that 3 1 , p and p 5 2 are the three consecutive terms of an AP Solution: We are given 3 1 , p, p 5 2 are in AP p p p ? ? ? 5 2 3 1 5 5 2 3 1 3 p p p ? ? ? ? 24 5 24 5 15 6 5 15 ? ? ? ? ? ? ? ? ? ? p p p p p Therefore, 24 5 ? p 10) Determine x, so that 7 6 , x and x 10 8 are the three consecutive terms of an AP Solution: We are given 7 6 , x, x 10 8 are in AP x x x ? ? ? 10 8 7 6 10 10 8 7 6 7 x x x ? ? ? ? 7 5 84 60 70 56 60 70 ? ? ? ? ? ? ? ? ? ? x x x x x Therefore, 7 5 ? x 11) Which term of the progression 15 , 5 1 14 , ,... 5 2 13 is the first negative term Arithmetic Progressions Page 5 1) Find 6 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 6 6 6 1 ? ? ? ? a 216 ) 1 ( 5 6 ? ? ? a = -216 2) Find 13 th term of the sequence defined by 3 1 ) 1 ( n a n n ? ? ? ? Solution: Here, 3 1 ) 1 ( n a n n ? ? ? ? ? ? 3 1 13 13 13 1 ? ? ? ? a 2197 ) 1 ( 12 13 ? ? ? a = 2197 3) The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term. Solution: Here, common difference (d) = -10 First term 200 1 ? a d n a a n ) 1 ( 1 ? ? ? ) 10 )( 1 20 ( 200 20 ? ? ? ? a 10 20 ? a 4) An arithmetic sequence has a common difference equal to 10 and its 6 th term is equal to 52. Find its 15 th term Solution: We use nth term formula d n a a n ) 1 ( 1 ? ? ? for the 6 th term ) 1 6 ( 10 52 1 6 6 ? ? ? ? a a a 50 52 1 ? ?a Therefore, 2 50 52 1 ? ? ? a Now, we know the first term and the common difference, we use the nth term formula to find the 15 th term. 142 ) 1 15 ( 10 2 15 ? ? ? ? a Arithmetic Progressions 5) If x times the x th term of an AP is equal to y times its y th term , Prove that the (x + y) th term of the AP is zero Solution: Let â€˜aâ€™ be the first term andâ€˜dâ€™ be the common difference of an AP By the given condition, y x yt xt ? ? ? ? ? d y a y d x a x ) 1 ( ) 1 ( ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ........ 0 ) 1 ( 0 ) 1 ( ) ( 0 ) ( ) )( ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?y x t x d y x a d y x a y x d y x y x y x a y x d y x y x a y x d y y d x x ya xa d y y ya d x x xa ? 6) Which term of AP 7, 14, 21,â€¦â€¦..is 133 Solution: Here, a = 7 , d = 14 â€“ 7= 7 Let 133 be the th n term of the AP ( Where N n ? ) 133 ? n t 133 ) 1 ( ? ? ? ? d n a ? ? N n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 19 . .......... 19 7 133 133 7 133 7 7 7 133 7 ) 1 ( 7 Therefore, 133 is the 19 th term of the given AP. Arithmetic Progressions 7) Which term of AP 10, 17, 24,â€¦â€¦..is 150 Solution: Here, a = 10 , d =17 â€“ 10 = 7 Let 150 be the th n term of the AP ( Where N n ? ) 150 ? n t n t d n a ? ? ? ? ) 1 ( ? ? N n n n n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 21 . .......... 21 7 147 147 7 150 7 3 150 7 7 10 150 7 ) 1 ( 10 Therefore, 150 is the 21 th term of the given AP. 8) An arithmetic sequence has its 5 th term equal to 22 and its 15 th term equal to 62. Find its 100 th term Solution: We use the formula for 5 th term and 15 th term ) 1 ......( .......... 4 22 ) 1 5 ( 1 1 5 d a d a a ? ? ? ? ? ) 2 ......( .......... 14 62 ) 1 15 ( 1 1 15 d a d a a ? ? ? ? ? Solving equation (1) and (2) we get value of d Therefore d = 4 Now, plug in the value of d in equation (1) we get the value of 1 a 6 16 22 16 22 1 1 1 ? ? ? ? ? ? ? a a a Now, we use nth term formula to find the 100 th term 402 4 ) 1 100 ( 6 100 100 ? ? ? ? a a Arithmetic Progressions 9) Determine p, so that 3 1 , p and p 5 2 are the three consecutive terms of an AP Solution: We are given 3 1 , p, p 5 2 are in AP p p p ? ? ? 5 2 3 1 5 5 2 3 1 3 p p p ? ? ? ? 24 5 24 5 15 6 5 15 ? ? ? ? ? ? ? ? ? ? p p p p p Therefore, 24 5 ? p 10) Determine x, so that 7 6 , x and x 10 8 are the three consecutive terms of an AP Solution: We are given 7 6 , x, x 10 8 are in AP x x x ? ? ? 10 8 7 6 10 10 8 7 6 7 x x x ? ? ? ? 7 5 84 60 70 56 60 70 ? ? ? ? ? ? ? ? ? ? x x x x x Therefore, 7 5 ? x 11) Which term of the progression 15 , 5 1 14 , ,... 5 2 13 is the first negative term Arithmetic Progressions Solution: The given expression is15, 5 1 14 , ,... 5 2 13 Here, 15 5 71 1 2 ? ? ?T T = 5 75 71 ? = 5 4 ? â€¦â€¦â€¦â€¦..(1) 5 71 5 67 2 3 ? ? ?T T = 5 4 ? Therefore, (1) is an AP with a = 15, 5 4 ? ? d Let the nth term of the given AP be the first negative term. Then, nth term < 0. ? 0 ? n T ? 0 5 4 ) 1 ( 15 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? n ? ? 0 4 79 ? ? ? n 79 4 ? ? n 4 3 19 ? ?n Therefore, n = 20, i.e. 20 th term is the first negative term in the given AP. 12) Which term of the progression 27, 5 2 26 , ,... 5 4 25 is the first negative term Solution: The given expression is 27 , 5 2 26 , ,... 5 4 25 Here, 27 5 132 1 2 ? ? ?T T = 5 135 132 ? Arithmetic ProgressionsRead More

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