Arithmetic Sequence and Nth Term Formula

Questions and Answers

What is the formula to find the nth term of an arithmetic sequence?

a_n = a_1 + (n - 1)d (correct)

a_n = a_1 * n + d

a_n = a_1 + (n + 1)d

a_n = n^2 + a_1 - d

When identifying the common difference in an arithmetic sequence, how is it calculated?

d = a_{n - 1} - a_n

d = a_n - a_{n - 1} (correct)

d = (a_n + a_{n - 1}) / 2

d = a_n + a_{n - 1}

Which formula can be used to find the sum of the first n terms of an arithmetic sequence if the nth term is unknown?

S_n = rac{2n}{5} (a_1 + a_n)

S_n = rac{n}{2} (2a_1 + nd) (correct)

S_n = n imes (a_1 + a_n)

S_n = n imes a_1 + (n - 1)d

In an arithmetic sequence, if the first term is 5 and the common difference is 3, what is the 4th term?

<p>17</p> Signup and view all the answers

What characteristic does the sum of an arithmetic sequence not possess?

<p>It can only be calculated if all terms are known.</p> Signup and view all the answers

Study Notes

Arithmetic Sequence

Definition: A sequence of numbers in which the difference between consecutive terms is constant.

Nth Term Formula

General form: ( a_n = a_1 + (n - 1)d )
- ( a_n ): nth term
- ( a_1 ): first term
- ( n ): term number
- ( d ): common difference

Common Difference

Definition: The fixed amount added or subtracted to get from one term to the next.
Calculation: ( d = a_{n} - a_{n - 1} )
Can be positive, negative, or zero.

Sum of Terms

Formula for the sum of the first n terms:
- ( S_n = \frac{n}{2} (a_1 + a_n) )
- or ( S_n = \frac{n}{2} (2a_1 + (n - 1)d) )
  - ( S_n ): sum of the first n terms
  - ( a_1 ): first term
  - ( a_n ): nth term
  - ( n ): number of terms
  - ( d ): common difference
Characteristics:
- The sum can be calculated even if the last term is unknown using the first term, common difference, and number of terms.

Arithmetic Sequence Overview

An arithmetic sequence features a constant difference between successive terms, forming a linear pattern.

Nth Term Formula

The formula for determining the nth term is expressed as ( a_n = a_1 + (n - 1)d ), where:
- ( a_n ) is the nth term of the sequence.
- ( a_1 ) represents the first term.
- ( n ) is the position in the sequence.
- ( d ) denotes the common difference.

Common Difference

The common difference ( d ) is defined as the constant value added or subtracted to arrive at the next term.
It is calculated using the formula ( d = a_{n} - a_{n - 1} ).
The common difference can take on positive, negative, or zero values, influencing the nature of the sequence.

Sum of Terms

The sum of the first n terms of an arithmetic sequence can be calculated with one of two formulas:
- ( S_n = \frac{n}{2} (a_1 + a_n) )
- Alternatively, ( S_n = \frac{n}{2} (2a_1 + (n - 1)d) ), where:
  - ( S_n ) is the sum of the first n terms.
  - ( a_1 ) is the initial term of the sequence.
  - ( a_n ) is the nth term.
  - ( n ) is the total number of terms considered.
  - ( d ) is the common difference.
This allows the sum to be calculated even when the last term is unknown, using the first term, common difference, and the number of terms.

Description

Test your knowledge of arithmetic sequences with this quiz. Learn about the definition, common difference, and how to calculate the nth term and sum of terms. This quiz will help solidify your understanding of these fundamental concepts in mathematics.