Certainly! Let's solve the problem step-by-step.
### Part (a): Finding the First Term of the Series
To find the first term of the arithmetic series, we need to evaluate the sum of the first term when [tex]\( n = 1 \)[/tex].
Given the sum of the first [tex]\( n \)[/tex] terms, [tex]\( S_n \)[/tex], is:
[tex]\[ S_n = n(2n + 3) \][/tex]
Substitute [tex]\( n = 1 \)[/tex] into the formula:
[tex]\[ S_1 = 1 \cdot (2 \cdot 1 + 3) \][/tex]
[tex]\[ S_1 = 1 \cdot (2 + 3) \][/tex]
[tex]\[ S_1 = 1 \cdot 5 \][/tex]
[tex]\[ S_1 = 5 \][/tex]
So, the first term of the series is [tex]\( 5 \)[/tex].
### Part (b): Finding the Common Difference of the Series
The common difference can be found by looking at the difference between the sum of the first two terms and the sum of the first term.
Firstly, evaluate [tex]\( S_2 \)[/tex], which is the sum of the first two terms, when [tex]\( n = 2 \)[/tex]:
[tex]\[ S_2 = 2 \cdot (2 \cdot 2 + 3) \][/tex]
[tex]\[ S_2 = 2 \cdot (4 + 3) \][/tex]
[tex]\[ S_2 = 2 \cdot 7 \][/tex]
[tex]\[ S_2 = 14 \][/tex]
We already know [tex]\( S_1 = 5 \)[/tex]. Given that [tex]\( S_2 \)[/tex] is the sum of the first two terms, then:
[tex]\[ S_2 = a_1 + a_2 \][/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( a_2 \)[/tex] is the second term.
So,
[tex]\[ 14 = 5 + a_2 \][/tex]
Solving for the second term, [tex]\( a_2 \)[/tex]:
[tex]\[ a_2 = 14 - 5 \][/tex]
[tex]\[ a_2 = 9 \][/tex]
Now, the common difference [tex]\( d \)[/tex] is the difference between the second term and the first term:
[tex]\[ d = a_2 - a_1 \][/tex]
[tex]\[ d = 9 - 5 \][/tex]
[tex]\[ d = 4 \][/tex]
Therefore, the common difference is [tex]\( 4 \)[/tex].
### Summary of Results
(a) The first term of the series is [tex]\( 5 \)[/tex].
(b) The common difference of the series is [tex]\( 4 \)[/tex].
