Understanding the Series
To find the nth term of a series given that the sum of the first n terms (S_n) is 3n, we start by establishing the relationship between the sum and the individual terms of the series.
Sum of the First n Terms
- The sum of the first n terms is defined as:
S_n = a_1 + a_2 + a_3 + ... + a_n
- Here, we know that S_n = 3n.
Finding the nth Term
- The nth term (a_n) can be found using the formula:
a_n = S_n - S_(n-1)
- To calculate S_(n-1), we can substitute (n-1) into the given sum formula:
S_(n-1) = 3(n - 1) = 3n - 3
- Now, we can derive the nth term:
a_n = S_n - S_(n-1)
a_n = 3n - (3n - 3)
a_n = 3
Conclusion
- The nth term of the series is constant:
- a_n = 3 for all n.
- This indicates that the series is an arithmetic series where every term is equal to 3.
Summary
- The sum of the first n terms is given as 3n.
- The nth term of the series is calculated as a_n = 3.
- The series consists of constant terms, each equal to 3.