Sure, let's find the first three terms of the arithmetic series given the first term [tex]\( a_1 = 3 \)[/tex], the [tex]\( n \)[/tex]-th term [tex]\( a_n = 24 \)[/tex], and the sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n = 108 \)[/tex].
We'll approach this problem step-by-step:
### Step 1: Find the number of terms [tex]\( n \)[/tex]
The sum of the first [tex]\( n \)[/tex] terms of an arithmetic series is given by the formula:
[tex]\[ S_n = \frac{n}{2} (a_1 + a_n) \][/tex]
Given:
[tex]\[ S_n = 108 \][/tex]
[tex]\[ a_1 = 3 \][/tex]
[tex]\[ a_n = 24 \][/tex]
We can substitute these values into the formula to solve for [tex]\( n \)[/tex]:
[tex]\[ 108 = \frac{n}{2} (3 + 24) \][/tex]
Simplify inside the parentheses:
[tex]\[ 108 = \frac{n}{2} \times 27 \][/tex]
To clear the fraction, multiply both sides by 2:
[tex]\[ 216 = 27n \][/tex]
Solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{216}{27} \][/tex]
[tex]\[ n = 8 \][/tex]
So, there are 8 terms in the arithmetic series.
### Step 2: Find the common difference [tex]\( d \)[/tex]
The [tex]\( n \)[/tex]-th term of an arithmetic series is given by the formula:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Given:
[tex]\[ a_n = 24 \][/tex]
[tex]\[ a_1 = 3 \][/tex]
[tex]\[ n = 8 \][/tex]
We can substitute these values into the formula to solve for [tex]\( d \)[/tex]:
[tex]\[ 24 = 3 + (8 - 1)d \][/tex]
Simplify the equation:
[tex]\[ 24 = 3 + 7d \][/tex]
Subtract 3 from both sides:
[tex]\[ 21 = 7d \][/tex]
Solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{21}{7} \][/tex]
[tex]\[ d = 3 \][/tex]
So, the common difference [tex]\( d \)[/tex] is 3.
### Step 3: Find the first three terms
The first term [tex]\( a_1 \)[/tex] is already given as 3.
To find the second term [tex]\( a_2 \)[/tex]:
[tex]\[ a_2 = a_1 + d \][/tex]
[tex]\[ a_2 = 3 + 3 \][/tex]
[tex]\[ a_2 = 6 \][/tex]
To find the third term [tex]\( a_3 \)[/tex]:
[tex]\[ a_3 = a_1 + 2d \][/tex]
[tex]\[ a_3 = 3 + 2 \times 3 \][/tex]
[tex]\[ a_3 = 3 + 6 \][/tex]
[tex]\[ a_3 = 9 \][/tex]
### Conclusion
The first three terms of the arithmetic series are:
1. [tex]\( a_1 = 3 \)[/tex]
2. [tex]\( a_2 = 6 \)[/tex]
3. [tex]\( a_3 = 9 \)[/tex]
