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: 1) For the Arithmetic Progression (AP) 25, 22, 19, ... with 22 terms: Step 1: Identify the first term (a), common difference (d), and number of terms (n). The first term is a = 25. The common difference is d = 22 - 25 = -3. The number of terms is n = 22. Step 2: Use the formula for the sum of an AP, S_n = (n)/(2)[2a + (n-1)d]. S_22 = (22)/(2)[2(25) + (22-1)(-3)] S_22 = 11[50 + (21)(-3)] S_22 = 11[50 - 63] S_22 = 11[-13] S_22 = -143 The sum of the AP is -143. 2) For an AP with 15 terms, common difference of -3, and a sum of 120: Step 1: Identify the given values and use the sum formula to find the first term (a). Given n = 15, d = -3, S_15 = 120. The formula for the sum of an AP is S_n = (n)/(2)[2a + (n-1)d]. 120 = (15)/(2)[2a + (15-1)(-3)] 120 = (15)/(2)[2a + (14)(-3)] 120 = (15)/(2)[2a - 42] Multiply both sides by (2)/(15): (120 × 2)/(15) = 2a - 42 16 = 2a - 42 16 + 42 = 2a 58 = 2a a = (58)/(2) a = 29 The first term is a = 29. Step 2: Use the formula for the n-th term of an AP, a_n = a + (n-1)d, to find the last term (a_15). a_15 = 29 + (15-1)(-3) a_15 = 29 + (14)(-3) a_15 = 29 - 42 a_15 = -13 The last term is -13. 3) For the sequence 2, 4, 8, 16, ... with 16 terms: Step 1: Identify the type of progression and its parameters. The sequence 2, 4, 8, 16, ... is a Geometric Progression (GP), not an AP, because each term is multiplied by a constant ratio to get the next term. The first term is a = 2. The common ratio is r = (4)/(2) = 2. The number of terms is n = 16. Step 2: Use the formula for the sum of a GP, S_n = (a(r^n - 1))/(r - 1) (since r > 1). S_16 = 2(2^16 - 1)2 - 1 S_16 = 2(2^16 - 1)1 S_16 = 2(65536 - 1) S_16 = 2(65535) S_16 = 131070 The sum of the GP is 131070. 4) For the Geometric Progression (GP) 8, 12, 18, ... with 6 terms: Step 1: Identify the first term (a), common ratio (r), and number of terms (n). The first term is a = 8. The common ratio is r = (12)/(8) = (3)/(2) = 1.5. The number of terms is n = 6. Step 2: Use the formula for the sum of a GP, S_n = (a(r^n - 1))/(r - 1) (since r > 1). S_6 = (8((3)/(2))^6 - 1)(3)/(2) - 1 S_6 = (8(3^6)/(2^6) - 1)(1)/(2) S_6 = 16((729)/(64) - 1) S_6 = 16((729 - 64)/(64)) S_6 = 16((665)/(64)) S_6 = (16 × 665)/(64) S_6 = (665)/(4) S_6 = 166.25 The sum of the GP is 166.25. Send me the next one 📸