AP Series and Terms Calculation

Finding the AP with Given Conditions

• The third term of the arithmetic progression (AP) is 16: (a + 2d = 16)
• The seventh term exceeds the fifth term by 12: (a + 6d - (a + 4d) = 12)
• Simplifying gives (2d = 12), leading to (d = 6)
• Substituting (d) back, (a + 2(6) = 16) yields (a = 4)
• The first few terms of the AP are: 4, 10, 16, 22...

Finding the 20th Term from a Given AP

• The first term (a = 253) and the common difference (d = 248 - 253 = -5)
• To find the 20th term, use the formula: (a_n = a + (n - 1) d)
• Calculation for the 20th term:
  • (a_{20} = 253 + (20 - 1)(-5))
  • Simplifying results in (a_{20} = 253 - 95 = 158)

Determining First Three Terms from Sum Conditions

• The sum of the 4th and 8th terms equals 24:
  • Equation: (a + 3d + a + 7d = 24)
  • This simplifies to (2a + 10d = 24) or (a + 5d = 12) (Equation 1)
• The sum of the 6th and 10th terms equals 44:
  • Equation: (a + 5d + a + 9d = 44)
  • This simplifies to (2a + 14d = 44) or (a + 7d = 22) (Equation 2)
• Solving both equations:
  • From Equation 1, substitute (d) into it.
  • Substituting (d) leads to (a + 5(5) = 12), resulting in (a + 25 = 12), thus (a = -13)
• The first three terms of the AP are: -13, -8, -3 (with (d = 5))