Formula to describe arithmetic sequence can be written as:
=> an = an – 1 + d, here 'a' is denotes the initial term and 'd' denotes the common difference between two consecutive terms.
If we want to find the explicit or closed form of an arithmetic sequence, then formula is given by:
=> an = a 1 + (n - 1) d, here an is n th term of the sequence. A1 is the first term in the sequence.
For example: Let the sequence of series 24, 28, 32, 36, 40,...... then write the explicit form of a sequence.
Solution: Given the sequence is 24, 28, 32, 36, 40,.....
First term in the sequence is 24 and common difference in each term is 28 – 24 = 4,
So value of d = 4. Now put these values in the formula. On putting these values in the formula we get:
=> an = a 1 + (n - 1) d,
=> an = 24 + (n - 1) 4, Now simplify the required expression to get the result.
=> an = 24 + 4n – 4, on further solving we get:
=> an = 4n + 20
So we get explicit or closed formula for the arithmetic sequence which is an = 4n + 20. This is how we write explicit form of a sequence.