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13. Without doing the actual addition, find the sumof :(i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19+ 21 + 23(ii) 1 + 3 + 5 + 7 + 9 + ................ +39 + 41(iii) 1 + 3 + 5+ 7 + 9 + ......... +51 + 53 |
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Answer» i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 Using Arithmetic Progression First term (a) = 1 Common difference (d) = 3 - 1 = 2 Last term (l) = 23 Number of terms (n) = 12 We have to find the summative (sum) of the GIVEN A.P. Sn = n/2 (a + l) → 12/2 (1 + 23) → 6 (24) → 144 ii) 1 + 3 + 5 + 7 + 9 + ................ +39 + 41 First term (a) = 1 Common difference = 3 - 1 = 2 Last term (an or l) = 41 an = a + (n - 1)d OR l = a + (n - 1)d 41 = 1 + (n - 1)2 40 = (n - 1)2 20 = n - 1 n = 21 Sn = n/2 (a + l) → 21/2 (1 + 41) → 21/2 (42) → 21(21) → 441 iii) 1 + 3 + 5+ 7 + 9 + ......... +51 + 53 First term (a) = 1 Common difference = 3 - 1 = 2 Last term (an or l) = 53 an = a + (n - 1)d OR l = a + (n - 1)d 53 = 1 + (n - 1)2 52 = (n - 1)2 26 = n - 1 27 = n Sn = n/2 (a + l) → 27/2 (1 + 53) → 27/2 (54) → 27(27) → 729 |
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