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Page 1 1. Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each. (i) (ii) 11, 6, 1, (iii) 5 2 1 1, , , ,.......... 6 3 2 (iv) 2, 8, 18, 32,.......... (v) 20, 45, 80, 125,......... Sol: (i) The given progression 9 Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 9 Common difference =6 Next term of the AP = 27 + 6 = 33 (ii) The given progression 11, 6, 1, Clearly,6 11 = 1 6 = 4 1 = 5 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 11 Common difference = 5 Next term of the 4 5 9 A P (iii) The given progression 5 2 1 1, , , ,.......... 6 3 2 Clearly, 5 2 5 1 2 1 1 6 3 6 2 3 6 (Constant) Thus, each term differs from its preceding term by 1 . 6 So, the given progression is an AP. First term = 1 Common difference = 1 6 Next tern of the 1 1 2 1 2 6 6 3 A P Page 2 1. Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each. (i) (ii) 11, 6, 1, (iii) 5 2 1 1, , , ,.......... 6 3 2 (iv) 2, 8, 18, 32,.......... (v) 20, 45, 80, 125,......... Sol: (i) The given progression 9 Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 9 Common difference =6 Next term of the AP = 27 + 6 = 33 (ii) The given progression 11, 6, 1, Clearly,6 11 = 1 6 = 4 1 = 5 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 11 Common difference = 5 Next term of the 4 5 9 A P (iii) The given progression 5 2 1 1, , , ,.......... 6 3 2 Clearly, 5 2 5 1 2 1 1 6 3 6 2 3 6 (Constant) Thus, each term differs from its preceding term by 1 . 6 So, the given progression is an AP. First term = 1 Common difference = 1 6 Next tern of the 1 1 2 1 2 6 6 3 A P (iv) The given progression 2, 8, 18, 32,.......... This sequence can be written as 2, 2 2,3 2,4 2,........ Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant) Thus, each term differs from its preceding term by 2, So, the given progression is an AP. First term 2 Common difference 2 Next tern of the 4 2 2 5 2 50 A P (v) This given progression 20, 45, 80, 125,......... This sequence can be re-written as 2 5,3 5, 4 5,5 5,........... Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant) Thus, each term differs from its preceding term by 5.So, the given progression is an AP. First term 2 5 20 Common difference 5 Next term of the 5 5 5 6 5 180 A P 2. Find: (i) the 20 th term of the AP 9,13,17,21,.......... (ii) the 35 th term of AP 20,17,14,11,.......... (iii) the 18 th term of the AP 2, 18, 50, 98,........... (iv) the 9 th term of the AP 3 5 7 9 , , , ,......... 4 4 4 4 (v) the 15the term of the AP 40, 15,10,35,......... Sol: (i) The given A P is 9,13,17,21,.......... First term, 9 a Common difference, 13 9 4 d t h n term of the AP, 1 9 1 4 n a a n d n 20 t h term of the AP, 20 9 20 1 4 9 76 85 a (ii) The given AP is 20,17,14,11,.......... First term, 20 a Common difference, 17 20 3 d t h n term of the AP, 1 20 1 3 n a a n d n Page 3 1. Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each. (i) (ii) 11, 6, 1, (iii) 5 2 1 1, , , ,.......... 6 3 2 (iv) 2, 8, 18, 32,.......... (v) 20, 45, 80, 125,......... Sol: (i) The given progression 9 Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 9 Common difference =6 Next term of the AP = 27 + 6 = 33 (ii) The given progression 11, 6, 1, Clearly,6 11 = 1 6 = 4 1 = 5 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 11 Common difference = 5 Next term of the 4 5 9 A P (iii) The given progression 5 2 1 1, , , ,.......... 6 3 2 Clearly, 5 2 5 1 2 1 1 6 3 6 2 3 6 (Constant) Thus, each term differs from its preceding term by 1 . 6 So, the given progression is an AP. First term = 1 Common difference = 1 6 Next tern of the 1 1 2 1 2 6 6 3 A P (iv) The given progression 2, 8, 18, 32,.......... This sequence can be written as 2, 2 2,3 2,4 2,........ Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant) Thus, each term differs from its preceding term by 2, So, the given progression is an AP. First term 2 Common difference 2 Next tern of the 4 2 2 5 2 50 A P (v) This given progression 20, 45, 80, 125,......... This sequence can be re-written as 2 5,3 5, 4 5,5 5,........... Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant) Thus, each term differs from its preceding term by 5.So, the given progression is an AP. First term 2 5 20 Common difference 5 Next term of the 5 5 5 6 5 180 A P 2. Find: (i) the 20 th term of the AP 9,13,17,21,.......... (ii) the 35 th term of AP 20,17,14,11,.......... (iii) the 18 th term of the AP 2, 18, 50, 98,........... (iv) the 9 th term of the AP 3 5 7 9 , , , ,......... 4 4 4 4 (v) the 15the term of the AP 40, 15,10,35,......... Sol: (i) The given A P is 9,13,17,21,.......... First term, 9 a Common difference, 13 9 4 d t h n term of the AP, 1 9 1 4 n a a n d n 20 t h term of the AP, 20 9 20 1 4 9 76 85 a (ii) The given AP is 20,17,14,11,.......... First term, 20 a Common difference, 17 20 3 d t h n term of the AP, 1 20 1 3 n a a n d n 35 t h term of the AP, 35 20 35 1 3 20 102 82 a (iii) The given AP is 2, 18, 50, 98,........... This can be re-written as 2,3 2,5 2,7 2,......... First term, 2 a Common difference, 3 2 2 2 2 d t h n term of the AP, 1 2 1 2 2 n a a n d n 18 t h term of the AP, 18 2 18 1 2 2 2 34 2 35 2 2450 a (iv) The given AP is 3 5 7 9 , , , ,......... 4 4 4 4 First term, 3 4 a Common difference, 5 3 2 1 4 4 4 2 d t h n term of the AP, 3 1 1 1 4 2 n a a n d n 9 t h t e r m of the AP, 9 3 1 3 19 9 1 4 4 2 4 4 a (v) The given AP is 40, 15,10,35,......... First term, 40 a Common difference, 15 40 25 d t h n term of the AP, 1 40 1 25 n a a n d n 15 t h term of the AP, 15 40 15 1 25 40 350 310 a 3. Find the 37 th term of the AP 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 Sol: The given AP is 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 First term, 6 a and common difference, 3 31 31 24 7 7 6 6 4 4 4 4 d Now, 37 37 1 36 T a d a d 7 6 36 6 63 69 4 37 t h term = 69 Page 4 1. Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each. (i) (ii) 11, 6, 1, (iii) 5 2 1 1, , , ,.......... 6 3 2 (iv) 2, 8, 18, 32,.......... (v) 20, 45, 80, 125,......... Sol: (i) The given progression 9 Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 9 Common difference =6 Next term of the AP = 27 + 6 = 33 (ii) The given progression 11, 6, 1, Clearly,6 11 = 1 6 = 4 1 = 5 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 11 Common difference = 5 Next term of the 4 5 9 A P (iii) The given progression 5 2 1 1, , , ,.......... 6 3 2 Clearly, 5 2 5 1 2 1 1 6 3 6 2 3 6 (Constant) Thus, each term differs from its preceding term by 1 . 6 So, the given progression is an AP. First term = 1 Common difference = 1 6 Next tern of the 1 1 2 1 2 6 6 3 A P (iv) The given progression 2, 8, 18, 32,.......... This sequence can be written as 2, 2 2,3 2,4 2,........ Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant) Thus, each term differs from its preceding term by 2, So, the given progression is an AP. First term 2 Common difference 2 Next tern of the 4 2 2 5 2 50 A P (v) This given progression 20, 45, 80, 125,......... This sequence can be re-written as 2 5,3 5, 4 5,5 5,........... Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant) Thus, each term differs from its preceding term by 5.So, the given progression is an AP. First term 2 5 20 Common difference 5 Next term of the 5 5 5 6 5 180 A P 2. Find: (i) the 20 th term of the AP 9,13,17,21,.......... (ii) the 35 th term of AP 20,17,14,11,.......... (iii) the 18 th term of the AP 2, 18, 50, 98,........... (iv) the 9 th term of the AP 3 5 7 9 , , , ,......... 4 4 4 4 (v) the 15the term of the AP 40, 15,10,35,......... Sol: (i) The given A P is 9,13,17,21,.......... First term, 9 a Common difference, 13 9 4 d t h n term of the AP, 1 9 1 4 n a a n d n 20 t h term of the AP, 20 9 20 1 4 9 76 85 a (ii) The given AP is 20,17,14,11,.......... First term, 20 a Common difference, 17 20 3 d t h n term of the AP, 1 20 1 3 n a a n d n 35 t h term of the AP, 35 20 35 1 3 20 102 82 a (iii) The given AP is 2, 18, 50, 98,........... This can be re-written as 2,3 2,5 2,7 2,......... First term, 2 a Common difference, 3 2 2 2 2 d t h n term of the AP, 1 2 1 2 2 n a a n d n 18 t h term of the AP, 18 2 18 1 2 2 2 34 2 35 2 2450 a (iv) The given AP is 3 5 7 9 , , , ,......... 4 4 4 4 First term, 3 4 a Common difference, 5 3 2 1 4 4 4 2 d t h n term of the AP, 3 1 1 1 4 2 n a a n d n 9 t h t e r m of the AP, 9 3 1 3 19 9 1 4 4 2 4 4 a (v) The given AP is 40, 15,10,35,......... First term, 40 a Common difference, 15 40 25 d t h n term of the AP, 1 40 1 25 n a a n d n 15 t h term of the AP, 15 40 15 1 25 40 350 310 a 3. Find the 37 th term of the AP 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 Sol: The given AP is 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 First term, 6 a and common difference, 3 31 31 24 7 7 6 6 4 4 4 4 d Now, 37 37 1 36 T a d a d 7 6 36 6 63 69 4 37 t h term = 69 4. Find the 25 th term of the AP 1 1 5,4 , 4,3 ,3,........ 2 2 Sol: The given AP is 1 1 5,4 , 4,3 ,3,........ 2 2 First term = 5 Common difference 1 9 9 10 1 4 5 5 2 2 2 2 1 5 2 a a n d d Now, 25 25 1 24 T a d a d 1 5 24 5 12 7 2 25 t h term 7 5. Find the nth term of each of the following Aps: (ii) 16, 9, 2, - Sol: (i) (6n 1) (ii) (23 7n) 6. If the nth term of a progression is (4n 10) show that it is an AP. Find its (i) first term, (ii) common difference (iii) 16 the term. Sol: 4 10 n n T [Given] 1 4 1 10 6 T 2 3 4 4 2 10 2 4 3 10 2 4 4 10 6 T T T Clearly, 2 6 2 2 6 2 4 (Constant) So, the terms 6, 2,2,6,...... forms an AP. Thus we have (i) First term 6 (ii) Common difference = 4 (iii) 16 1 15 6 15 4 54 T a n d a d Page 5 1. Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each. (i) (ii) 11, 6, 1, (iii) 5 2 1 1, , , ,.......... 6 3 2 (iv) 2, 8, 18, 32,.......... (v) 20, 45, 80, 125,......... Sol: (i) The given progression 9 Clearly, 15 9 = 21 15 = 27 21 = 6 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 9 Common difference =6 Next term of the AP = 27 + 6 = 33 (ii) The given progression 11, 6, 1, Clearly,6 11 = 1 6 = 4 1 = 5 (Constant) Thus, each term differs from its preceding term by 6. So, the given progression is an AP. First term = 11 Common difference = 5 Next term of the 4 5 9 A P (iii) The given progression 5 2 1 1, , , ,.......... 6 3 2 Clearly, 5 2 5 1 2 1 1 6 3 6 2 3 6 (Constant) Thus, each term differs from its preceding term by 1 . 6 So, the given progression is an AP. First term = 1 Common difference = 1 6 Next tern of the 1 1 2 1 2 6 6 3 A P (iv) The given progression 2, 8, 18, 32,.......... This sequence can be written as 2, 2 2,3 2,4 2,........ Clearly, 2 2 2 3 2 2 2 4 2 3 2 2 (Constant) Thus, each term differs from its preceding term by 2, So, the given progression is an AP. First term 2 Common difference 2 Next tern of the 4 2 2 5 2 50 A P (v) This given progression 20, 45, 80, 125,......... This sequence can be re-written as 2 5,3 5, 4 5,5 5,........... Clearly, 3 5 2 5 4 5 3 5 5 5 4 5 5 (Constant) Thus, each term differs from its preceding term by 5.So, the given progression is an AP. First term 2 5 20 Common difference 5 Next term of the 5 5 5 6 5 180 A P 2. Find: (i) the 20 th term of the AP 9,13,17,21,.......... (ii) the 35 th term of AP 20,17,14,11,.......... (iii) the 18 th term of the AP 2, 18, 50, 98,........... (iv) the 9 th term of the AP 3 5 7 9 , , , ,......... 4 4 4 4 (v) the 15the term of the AP 40, 15,10,35,......... Sol: (i) The given A P is 9,13,17,21,.......... First term, 9 a Common difference, 13 9 4 d t h n term of the AP, 1 9 1 4 n a a n d n 20 t h term of the AP, 20 9 20 1 4 9 76 85 a (ii) The given AP is 20,17,14,11,.......... First term, 20 a Common difference, 17 20 3 d t h n term of the AP, 1 20 1 3 n a a n d n 35 t h term of the AP, 35 20 35 1 3 20 102 82 a (iii) The given AP is 2, 18, 50, 98,........... This can be re-written as 2,3 2,5 2,7 2,......... First term, 2 a Common difference, 3 2 2 2 2 d t h n term of the AP, 1 2 1 2 2 n a a n d n 18 t h term of the AP, 18 2 18 1 2 2 2 34 2 35 2 2450 a (iv) The given AP is 3 5 7 9 , , , ,......... 4 4 4 4 First term, 3 4 a Common difference, 5 3 2 1 4 4 4 2 d t h n term of the AP, 3 1 1 1 4 2 n a a n d n 9 t h t e r m of the AP, 9 3 1 3 19 9 1 4 4 2 4 4 a (v) The given AP is 40, 15,10,35,......... First term, 40 a Common difference, 15 40 25 d t h n term of the AP, 1 40 1 25 n a a n d n 15 t h term of the AP, 15 40 15 1 25 40 350 310 a 3. Find the 37 th term of the AP 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 Sol: The given AP is 3 1 1 6,7 ,9 ,11 ,......... 4 2 4 First term, 6 a and common difference, 3 31 31 24 7 7 6 6 4 4 4 4 d Now, 37 37 1 36 T a d a d 7 6 36 6 63 69 4 37 t h term = 69 4. Find the 25 th term of the AP 1 1 5,4 , 4,3 ,3,........ 2 2 Sol: The given AP is 1 1 5,4 , 4,3 ,3,........ 2 2 First term = 5 Common difference 1 9 9 10 1 4 5 5 2 2 2 2 1 5 2 a a n d d Now, 25 25 1 24 T a d a d 1 5 24 5 12 7 2 25 t h term 7 5. Find the nth term of each of the following Aps: (ii) 16, 9, 2, - Sol: (i) (6n 1) (ii) (23 7n) 6. If the nth term of a progression is (4n 10) show that it is an AP. Find its (i) first term, (ii) common difference (iii) 16 the term. Sol: 4 10 n n T [Given] 1 4 1 10 6 T 2 3 4 4 2 10 2 4 3 10 2 4 4 10 6 T T T Clearly, 2 6 2 2 6 2 4 (Constant) So, the terms 6, 2,2,6,...... forms an AP. Thus we have (i) First term 6 (ii) Common difference = 4 (iii) 16 1 15 6 15 4 54 T a n d a d 7. Sol: In the given AP, 6 a and 10 6 4 d Suppose that there are n terms in the given AP. Then, 174 n T 1 174 a n d 6 1 4 174 2 4 174 4 172 43 n n n n Hence, there are 43 terms in the given AP. 8. How many terms are there in the AP 41, 3 Sol: In the given AP, 41 38 41 3 a a n d d Suppose that there are n terms in the given AP. Then 8 n T 1 8 41 1 3 8 44 3 8 3 36 12 a n d n n n n Hence, there are 12 terms in the given AP. 9. How many terms are there in the AP 1 18,15 ,13,....., 47. 2 ? Sol: The given AP is 1 18,15 ,13,....., 47. 2 First term, 18 a Common difference, 1 31 31 36 5 15 18 18 2 2 2 2 d Suppose there re n terms in the given AP. Then,Read More
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