Sequence and Series

Arithmetic Sequence

An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). 

Formula for the nth term of an arithmetic sequence: 

Where: 

- \( t_n \) = nth term 

- \( a \) = first term 

- \( d \) = common difference 

- \( n \) = term number 

Example: 

Find the 10th term of the sequence: 2, 5, 8, 11, ... 

- Here, \( a = 2 \), \( d = 3 \), and \( n = 10 \). 

- Using the formula: 

  \( t_{10} = 2 + (10 - 1) \times 3 = 2 + 27 = 29\)

- Answer: The 10th term is 29. 

Means of an Arithmetic Sequence

The arithmetic means represented by \(m_1, m_2, m_3, ……. m_n\) are the values between the first and the last terms in an arithmetic sequence. 

Formula for the arithmetic mean:

1. If there are two first(a) and last term(b):

\(M = \frac{a + b}{2}\) 

Where \( a \) and \( b \) are two terms in the sequence. 

Example: 

Q. Find the arithmetic mean between 4 and 10. 

\(M = \frac{4 + 10}{2} = \frac{14}{2} = 7\) 

- Answer: The arithmetic mean is 7.