Arithmetic progressions (APs) form a fundamental part of mathematics, featuring in various applications from simple calculations to complex problem solving. Understanding APs involves grasping the core concepts of the nth term, the sum of the first n terms, and their characteristic properties.

Definition and Basic Concepts

• Arithmetic Progression (AP): A sequence of numbers in which the difference of any two successive members is a constant. For instance, the sequence 1, 3, 5, 7, 9, 11... is an AP where each term increases by 2.
• Common Difference: Denoted as 'd', it is the fixed difference between two consecutive terms in the sequence.
• First Term: Denoted as 'a', it is the initial term of the sequence.

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Formulae

• Nth Term of an AP: The nth term (denoted as $u_n$) is calculated using the formula: $u_n = a + (n - 1)d$

where 'n' is the term number.

• Sum of the First n Terms: The sum of the first n terms (denoted as $S_n$) can be calculated using either of the following formulas:

$S_n = \frac{1}{2} n [2a + (n - 1)d]$

or

$S_n = \frac{n}{2} (a + l)$

where 'l' is the last term of the sequence.

Characteristic Property of an Arithmetic Progression

In an arithmetic progression, the sum of any two non-consecutive terms is equal to twice the term exactly in the middle of these two terms. Mathematically, this is represented as:

$2b = a + c$

where 'a' and 'c' are any non-consecutive terms and 'b' is the term exactly between them in the sequence.

Examples

Example 1: Finding a Specific Term

Question: Find the 120th term of the arithmetic sequence: 2, 4, 6, 8, 10...

Solution:

• First term (a): 2
• Common difference (d): 2
• Nth term formula: $u_{120} = 2 + (120 - 1) \times 2$
• Calculation: $u_{120} = 2 + 238 = 240$

Answer: The 120th term is 240.

Example 2: Finding the Sum of Terms

Question: Find the sum of the first 20 terms of the arithmetic progression with its first term being 7 and its 8th term being 28.

Solution:

• First term (a): 7
• Finding common difference (d):
  • $u_8 = 28 = 7 + (8 - 1)d$
  • $d = \frac{28 - 7}{7} = 3$
• Sum formula: $S_{20} = \frac{20}{2} [2 \times 7 + (20 - 1) \times 3]$
• Calculation: $S_{20} = 10 \times (14 + 57) = 710$

Answer: The sum of the first 20 terms is 710.