AP Series and Terms Calculation

What is the first term of the arithmetic progression (AP) where the 3rd term is 16 and the 7th term exceeds the 5th term by 12?

Calculate the 20th term from the last term of the AP: 253, 248, 243...?

If the sum of the 4th and 8th term of an AP is 24, what equation can be formed to express this relationship?

What are the first three terms of the AP found if the first term 'a' is -13 and the common difference is 5?

Determine the common difference 'd' in the AP where 2d equals 12.

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))

Test your knowledge on arithmetic progressions (AP) by solving problems related to finding terms and their values. This quiz covers finding specific terms of an AP, calculating differences between terms, and deriving formulas related to APs. Sharpen your skills in sequences and series!