This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Exercise 7.1: (1) Find the sums of the following A.P.s as far as the terms indicate. a) 5 + 10 + 15 + + 12th term Step 1: Identify the first term (a), common difference (d), and number of terms (n). a = 5 d = 10 - 5 = 5 n = 12 Step 2: Use the formula for the sum of an A.P., S_n = (n)/(2)(2a + (n-1)d). S_12 = (12)/(2)(2(5) + (12-1)5) S_12 = 6(10 + (11)5) S_12 = 6(10 + 55) S_12 = 6(65) S_12 = 390 The sum of the first 12 terms is 390. b) 1(1)/(4) + 1 + + nth term Step 1: Identify the first term (a), common difference (d), and number of terms (n). a = 1(1)/(4) = (5)/(4) d = 1 - (5)/(4) = -(1)/(4) n = n Step 2: Use the formula for the sum of an A.P., S_n = (n)/(2)(2a + (n-1)d). S_n = (n)/(2)(2((5)/(4)) + (n-1)(-(1)/(4))) S_n = (n)/(2)((10)/(4) - (n-1)/(4)) S_n = (n)/(2)((10 - (n-1))/(4)) S_n = (n)/(2)((10 - n + 1)/(4)) S_n = (n)/(2)((11 - n)/(4)) S_n = (n(11 - n))/(8) The sum of the first n terms is (n(11 - n))/(8). c) 1 + 2 + 3 + 4 + + 50th term Step 1: Identify the first term (a), common difference (d), and number of terms (n). a = 1 d = 2 - 1 = 1 n = 50 Step 2: Use the formula for the sum of an A.P., S_n = (n)/(2)(2a + (n-1)d). S_50 = (50)/(2)(2(1) + (50-1)1) S_50 = 25(2 + 49) S_50 = 25(51) S_50 = 1275 The sum of the first 50 terms is 1275. (2) The second term of an A.P is 15, and the 5th term is 21. Find the common difference, the first term and the sum of the first ten terms. Step 1: Write the given information using the formula for the n-th term of an A.P., T_n = a + (n-1)d. T_2 = a + (2-1)d a + d = 15 (1) T_5 = a + (5-1)d a + 4d = 21 (2) Step 2: Subtract equation (1) from equation (2) to find the common difference (d). (a + 4d) - (a + d) = 21 - 15 3d = 6 d = (6)/(3) d = 2 The common difference is 2. Step 3: Substitute d=2 into equation (1) to find the first term (a). a + 2 = 15 a = 15 - 2 a = 13 The first term is 13. Step 4: Find the sum of the first ten terms (S_10) using S_n = (n)/(2)(2a + (n-1)d). S_10 = (10)/(2)(2(13) + (10-1)2) S_10 = 5(26 + (9)2) S_10 = 5(26 + 18) S_10 = 5(44) S_10 = 220 The sum of the first ten terms is 220. (3) Find the number of terms in the following A.P.s a) 2 + 4 + + 46 Step 1: Identify the first term (a), common difference (d), and the last term (T_n). a = 2 d = 4 - 2 = 2 T_n = 46 Step 2: Use the formula for the n-th term of an A.P., T_n = a + (n-1)d, to find n. 46 = 2 + (n-1)2 46 - 2 = (n-1)2 44 = 2(n-1) (44)/(2) = n-1 22 = n-1 n = 22 + 1 n = 23 The number of terms is 23. b) 2.7 + 3.2 + + 17.7 Step 1: Identify the first term (a), common difference (d), and the last term (T_n). a = 2.7 d = 3.2 - 2.7 = 0.5 T_n = 17.7 Step 2: Use the formula for the n-th term of an A.P., T_n = a + (n-1)d, to find n. 17.7 = 2.7 + (n-1)0.5 17.7 - 2.7 = (n-1)0.5 15 = (n-1)0.5 (15)/(0.5) = n-1 30 = n-1 n = 30 + 1 n = 31 The number of terms is 31. (4) Prove that the sum of the integers from 1 to n is (1)/(2)n(n+1) = (n(n+1))/(2). Step 1: The sum of the integers from 1 to n is an arithmetic progression with: First term (a) = 1 Common difference (d) = 1 Number of terms (n) = n Last term (L) = n Step 2: Use the formula for the sum of an A.P., S_n = (n)/(2)(a + L). S_n = (n)/(2)(1 + n) S_n = (n(n+1))/(2) This proves that the sum of the integers from 1 to n is (n(n+1))/(2).