Arithmetic Progression Definition and Properties

Questions and Answers

What is the formula for the n-th term of an arithmetic progression?

$a_n = a + d^{(n - 1)}$

$a_n = a + n imes d$

$a_n = a + d/n$

$a_n = a + (n - 1)d$ (correct)

If the first term of an arithmetic progression is 5 and the common difference is 3, what is the 10th term?

32 (correct)

What is the sum of the first n terms of an arithmetic progression when n is known?

$S_n = n imes (a + d)$

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

$S_n = n imes a_n$

$S_n = a + n imes d$

In an arithmetic progression, if the common difference is 0, what can be said about the terms?

<p>All terms are equal to the first term.</p> Signup and view all the answers

If the average of the first and last terms of an arithmetic progression is equal to the average of all its terms, what can be concluded?

<p>The sequence is an arithmetic progression.</p> Signup and view all the answers

Study Notes

Definition and Properties

Arithmetic Progression (AP): A sequence of numbers in which the difference between consecutive terms is constant.
- General form: ( a, a + d, a + 2d, a + 3d, \ldots )
- Where:
  - ( a ) = first term
  - ( d ) = common difference
Properties:
- The ( n )-th term can be expressed as ( a_n = a + (n - 1)d ).
- The terms are uniformly spaced; thus, the average of the first and last term is equal to the average of all terms.
- If ( n ) is odd, the middle term is the average of the first and last term.
- The sum of the first ( n ) terms can be calculated using specific formulas.

Formulas For Nth Term

Nth Term Formula:
- ( a_n = a + (n - 1)d )
  - ( a_n ) = n-th term
  - ( a ) = first term
  - ( d ) = common difference
  - ( n ) = term number

Sum Of First N Terms

Sum Formula:
- ( S_n = \frac{n}{2} [2a + (n - 1)d] ) (when ( n ) is known)
- Alternatively: ( S_n = \frac{n}{2} [a + a_n] ) (using the first and n-th term)
  - ( S_n ) = sum of the first ( n ) terms
  - ( a ) = first term
  - ( a_n ) = n-th term
  - ( d ) = common difference
  - ( n ) = number of terms
Special Cases:
- If ( d = 0 ) (constant sequence), each term is equal to ( a ), and ( S_n = n \cdot a ).
- If ( n ) is large, the sum can be approximated in different contexts (e.g., for series analysis).

Arithmetic Progression (AP)

A sequence where the difference between consecutive terms remains constant.
General form: ( a, a + d, a + 2d, a + 3d, \ldots )
- ( a ): first term
- ( d ): common difference

Key Properties of AP

The ( n )-th term can be calculated using ( a_n = a + (n - 1)d ).
Terms are uniformly spaced, making the average of the first and last term equal to the average of all terms in the sequence.
When ( n ) is odd, the middle term represents the average of the first and last terms.
The sum of the first ( n ) terms can be determined using specific formulas.

Nth Term Formula

Formula for the ( n )-th term: ( a_n = a + (n - 1)d )
- ( a_n ): value of the n-th term
- ( a ): first term of the sequence
- ( d ): common difference
- ( n ): position of the term in the sequence

Sum of First N Terms

Sum formula when ( n ) is given: ( S_n = \frac{n}{2} [2a + (n - 1)d] ).
An alternative sum formula is ( S_n = \frac{n}{2} [a + a_n] ), utilizing the first and n-th terms.
- ( S_n ): total of the first ( n ) terms
- ( a ): first term
- ( a_n ): value of the n-th term
- ( d ): common difference
- ( n ): total number of terms

Special Cases in AP

If ( d = 0 ), all terms in the sequence are equal to ( a ), resulting in ( S_n = n \cdot a ).
For large ( n ), the sum may be approximated depending on the context (e.g., in convergence analysis or series).

Description

This quiz covers the essential definitions and properties of Arithmetic Progression (AP). It includes the general form, formulas for the n-th term, and the sum of the first n terms. Test your understanding of these concepts and their applications.