Page 1 1. Find the sum of each of the following Aps: (iii) -37, -33, - (iv) 1 1 1 , , ,..... 15 12 10 to 11 terms. Sol: (i) Here, 2 a and 7 2 5 d Using the formula, 2 1 , 2 n n S a n d we have 19 19 2 2 19 1 5 2 19 4 90 2 19 94 2 893 S (ii) Here, 9 a and 7 9 2 d Using the formula, 2 1 , 2 n n S a n d we have 14 14 2 9 14 1 2 2 7 18 26 7 8 56 S (iii) The given AP is -37, -33, - Here, 37 a and 33 37 33 37 4 d Using the formula, 2 1 , 2 n n S a n d we have 12 12 2 37 12 1 4 2 S 6 74 44 6 30 Page 2 1. Find the sum of each of the following Aps: (iii) -37, -33, - (iv) 1 1 1 , , ,..... 15 12 10 to 11 terms. Sol: (i) Here, 2 a and 7 2 5 d Using the formula, 2 1 , 2 n n S a n d we have 19 19 2 2 19 1 5 2 19 4 90 2 19 94 2 893 S (ii) Here, 9 a and 7 9 2 d Using the formula, 2 1 , 2 n n S a n d we have 14 14 2 9 14 1 2 2 7 18 26 7 8 56 S (iii) The given AP is -37, -33, - Here, 37 a and 33 37 33 37 4 d Using the formula, 2 1 , 2 n n S a n d we have 12 12 2 37 12 1 4 2 S 6 74 44 6 30 180 (iv) The given AP is 1 1 1 , , ,...... 15 12 10 Here, 1 15 a and 1 1 5 4 1 12 15 60 60 d Using the formula, 2 1 , 2 n n S a n d we have 11 11 1 1 2 11 1 2 15 60 11 2 10 2 15 60 S 11 18 2 60 33 20 (v) The given AP is 0.6,1.7,2.8,.......... Here, 0.6 a and 1.7 0.6 1.1 d Using formula, 2 1 , 2 n n S a n d we have 100 100 2 0.6 100 1 1.1 2 S 50 1.2 108.9 50 110.1 5505 2. Find the sum of each of the following arithmetic series: (i) 1 7 10 14 ... 84 2 (ii) 34 32 30 ... 10 (iii) ( 5) ( 8) ( 11) ... ( 230) Sol: (i) The given arithmetic series is 1 7 10 14 ..... 84. 2 Here, 1 21 21 4 7 7, 10 7 7 2 2 2 2 a d and 84. l Let the given series contains n terms. Then, Page 3 1. Find the sum of each of the following Aps: (iii) -37, -33, - (iv) 1 1 1 , , ,..... 15 12 10 to 11 terms. Sol: (i) Here, 2 a and 7 2 5 d Using the formula, 2 1 , 2 n n S a n d we have 19 19 2 2 19 1 5 2 19 4 90 2 19 94 2 893 S (ii) Here, 9 a and 7 9 2 d Using the formula, 2 1 , 2 n n S a n d we have 14 14 2 9 14 1 2 2 7 18 26 7 8 56 S (iii) The given AP is -37, -33, - Here, 37 a and 33 37 33 37 4 d Using the formula, 2 1 , 2 n n S a n d we have 12 12 2 37 12 1 4 2 S 6 74 44 6 30 180 (iv) The given AP is 1 1 1 , , ,...... 15 12 10 Here, 1 15 a and 1 1 5 4 1 12 15 60 60 d Using the formula, 2 1 , 2 n n S a n d we have 11 11 1 1 2 11 1 2 15 60 11 2 10 2 15 60 S 11 18 2 60 33 20 (v) The given AP is 0.6,1.7,2.8,.......... Here, 0.6 a and 1.7 0.6 1.1 d Using formula, 2 1 , 2 n n S a n d we have 100 100 2 0.6 100 1 1.1 2 S 50 1.2 108.9 50 110.1 5505 2. Find the sum of each of the following arithmetic series: (i) 1 7 10 14 ... 84 2 (ii) 34 32 30 ... 10 (iii) ( 5) ( 8) ( 11) ... ( 230) Sol: (i) The given arithmetic series is 1 7 10 14 ..... 84. 2 Here, 1 21 21 4 7 7, 10 7 7 2 2 2 2 a d and 84. l Let the given series contains n terms. Then, 84 7 7 1 84 1 2 7 7 84 2 2 7 7 161 84 2 2 2 161 23 7 n n a n a a n d n n n Required sum 23 7 84 2 2 n n S a l 23 91 2 2030 2 1 1046 2 (ii) The given arithmetic series is 34 32 30 ..... 10. Here, 34, 32 34 2 a d and 10. l Let the given series contain n terms. Then, 10 34 1 2 10 1 2 36 10 2 10 36 26 13 n n a n a a n d n n n Required sum 13 34 10 2 2 n n S a l 13 44 2 286 (iii) The given arithmetic series is 5 8 11 ....... 230 . Here, 5, 8 5 8 5 3 a d and 230. l Let the given series contain n terms. Then, Page 4 1. Find the sum of each of the following Aps: (iii) -37, -33, - (iv) 1 1 1 , , ,..... 15 12 10 to 11 terms. Sol: (i) Here, 2 a and 7 2 5 d Using the formula, 2 1 , 2 n n S a n d we have 19 19 2 2 19 1 5 2 19 4 90 2 19 94 2 893 S (ii) Here, 9 a and 7 9 2 d Using the formula, 2 1 , 2 n n S a n d we have 14 14 2 9 14 1 2 2 7 18 26 7 8 56 S (iii) The given AP is -37, -33, - Here, 37 a and 33 37 33 37 4 d Using the formula, 2 1 , 2 n n S a n d we have 12 12 2 37 12 1 4 2 S 6 74 44 6 30 180 (iv) The given AP is 1 1 1 , , ,...... 15 12 10 Here, 1 15 a and 1 1 5 4 1 12 15 60 60 d Using the formula, 2 1 , 2 n n S a n d we have 11 11 1 1 2 11 1 2 15 60 11 2 10 2 15 60 S 11 18 2 60 33 20 (v) The given AP is 0.6,1.7,2.8,.......... Here, 0.6 a and 1.7 0.6 1.1 d Using formula, 2 1 , 2 n n S a n d we have 100 100 2 0.6 100 1 1.1 2 S 50 1.2 108.9 50 110.1 5505 2. Find the sum of each of the following arithmetic series: (i) 1 7 10 14 ... 84 2 (ii) 34 32 30 ... 10 (iii) ( 5) ( 8) ( 11) ... ( 230) Sol: (i) The given arithmetic series is 1 7 10 14 ..... 84. 2 Here, 1 21 21 4 7 7, 10 7 7 2 2 2 2 a d and 84. l Let the given series contains n terms. Then, 84 7 7 1 84 1 2 7 7 84 2 2 7 7 161 84 2 2 2 161 23 7 n n a n a a n d n n n Required sum 23 7 84 2 2 n n S a l 23 91 2 2030 2 1 1046 2 (ii) The given arithmetic series is 34 32 30 ..... 10. Here, 34, 32 34 2 a d and 10. l Let the given series contain n terms. Then, 10 34 1 2 10 1 2 36 10 2 10 36 26 13 n n a n a a n d n n n Required sum 13 34 10 2 2 n n S a l 13 44 2 286 (iii) The given arithmetic series is 5 8 11 ....... 230 . Here, 5, 8 5 8 5 3 a d and 230. l Let the given series contain n terms. Then, 230 5 1 3 230 1 3 2 230 3 230 2 228 76 n n a n a a n d n n n Required sum 76 5 230 2 2 n n S a l 76 235 2 8930 3. Find the sum of first n terms of an AP whose nth term is (5 - 6n). Hence, find the sum of its first 20 terms. Sol: Let n a be the nth term of the AP. 5 6 n a n Putting 1, n we get First term, 1 5 6 1 1 a a Putting 2, n we get 2 5 6 2 7 a Let d be the common difference of the AP. 2 1 7 ( 1) 7 1 6 d a a Sum of first n tern of the AP, n S 2 2 1 1 6 2 1 2 2 2 6 6 2 2 3 2 3 n n n n S a n d n n n n n n Putting 20, n we get 2 20 2 20 3 20 40 1200 1160 S 4. The sum of the first n terms of an AP is 2 3 6 . n n Find the nth term and the 15 th term of this AP. Sol: Page 5 1. Find the sum of each of the following Aps: (iii) -37, -33, - (iv) 1 1 1 , , ,..... 15 12 10 to 11 terms. Sol: (i) Here, 2 a and 7 2 5 d Using the formula, 2 1 , 2 n n S a n d we have 19 19 2 2 19 1 5 2 19 4 90 2 19 94 2 893 S (ii) Here, 9 a and 7 9 2 d Using the formula, 2 1 , 2 n n S a n d we have 14 14 2 9 14 1 2 2 7 18 26 7 8 56 S (iii) The given AP is -37, -33, - Here, 37 a and 33 37 33 37 4 d Using the formula, 2 1 , 2 n n S a n d we have 12 12 2 37 12 1 4 2 S 6 74 44 6 30 180 (iv) The given AP is 1 1 1 , , ,...... 15 12 10 Here, 1 15 a and 1 1 5 4 1 12 15 60 60 d Using the formula, 2 1 , 2 n n S a n d we have 11 11 1 1 2 11 1 2 15 60 11 2 10 2 15 60 S 11 18 2 60 33 20 (v) The given AP is 0.6,1.7,2.8,.......... Here, 0.6 a and 1.7 0.6 1.1 d Using formula, 2 1 , 2 n n S a n d we have 100 100 2 0.6 100 1 1.1 2 S 50 1.2 108.9 50 110.1 5505 2. Find the sum of each of the following arithmetic series: (i) 1 7 10 14 ... 84 2 (ii) 34 32 30 ... 10 (iii) ( 5) ( 8) ( 11) ... ( 230) Sol: (i) The given arithmetic series is 1 7 10 14 ..... 84. 2 Here, 1 21 21 4 7 7, 10 7 7 2 2 2 2 a d and 84. l Let the given series contains n terms. Then, 84 7 7 1 84 1 2 7 7 84 2 2 7 7 161 84 2 2 2 161 23 7 n n a n a a n d n n n Required sum 23 7 84 2 2 n n S a l 23 91 2 2030 2 1 1046 2 (ii) The given arithmetic series is 34 32 30 ..... 10. Here, 34, 32 34 2 a d and 10. l Let the given series contain n terms. Then, 10 34 1 2 10 1 2 36 10 2 10 36 26 13 n n a n a a n d n n n Required sum 13 34 10 2 2 n n S a l 13 44 2 286 (iii) The given arithmetic series is 5 8 11 ....... 230 . Here, 5, 8 5 8 5 3 a d and 230. l Let the given series contain n terms. Then, 230 5 1 3 230 1 3 2 230 3 230 2 228 76 n n a n a a n d n n n Required sum 76 5 230 2 2 n n S a l 76 235 2 8930 3. Find the sum of first n terms of an AP whose nth term is (5 - 6n). Hence, find the sum of its first 20 terms. Sol: Let n a be the nth term of the AP. 5 6 n a n Putting 1, n we get First term, 1 5 6 1 1 a a Putting 2, n we get 2 5 6 2 7 a Let d be the common difference of the AP. 2 1 7 ( 1) 7 1 6 d a a Sum of first n tern of the AP, n S 2 2 1 1 6 2 1 2 2 2 6 6 2 2 3 2 3 n n n n S a n d n n n n n n Putting 20, n we get 2 20 2 20 3 20 40 1200 1160 S 4. The sum of the first n terms of an AP is 2 3 6 . n n Find the nth term and the 15 th term of this AP. Sol: Let n S denotes the sum of first n terms of the AP. 2 2 1 2 2 3 6 3 1 6 1 3 2 1 6 1 3 3 n n S n n S n n n n n n t h n term of the AP, n a 1 2 2 3 6 3 3 6 3 n n S S n n n n Putting 15, n we get 15 6 15 3 90 3 93 a Hence, the t h n term is 6 3 n and 15 th term is 93. 5. The sum of the first n terms of an AP is given by 2 3 n S n n . Find its (i) nth term, (ii) first term and (iii) common difference. Sol: Given: 2 3 ...... n S n n i Replacing n by 1 n in (i), we get: 2 1 2 2 3 1 1 3 2 1 1 3 7 4 n S n n n n n n n (i) Now, 1 n n n T S S 2 2 3 3 7 4 6 4 n n n n n t h n term, 6 4 ....... n T n i i (ii) Putting 1 n in (ii), we get: 1 6 1 4 2 T (iii) Putting 2 n in (ii), we get: 2 6 2 4 8 T Common difference, 2 1 8 2 6 d T TRead More

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