To find the sum of the first 25 terms of the given arithmetic sequence, follow these steps:
1. Identify the first term (a) and the common difference (d):
- The first term [tex]\( a \)[/tex] is the first number in the sequence. So, [tex]\( a = 2 \)[/tex].
- The common difference [tex]\( d \)[/tex] is the difference between two consecutive terms. Here, it's [tex]\( 8 - 2 = 6 \)[/tex].
2. Use the formula for the sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence:
The formula for the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\][/tex]
where [tex]\( n \)[/tex] is the number of terms, [tex]\( a \)[/tex] is the first term, and [tex]\( d \)[/tex] is the common difference.
3. Substitute the known values into the formula:
- [tex]\( a = 2 \)[/tex]
- [tex]\( d = 6 \)[/tex]
- [tex]\( n = 25 \)[/tex]
So, we substitute these values into the formula:
[tex]\[
S_{25} = \frac{25}{2} \times (2 \times 2 + (25-1) \times 6)
\][/tex]
4. Simplify inside the parentheses first:
[tex]\[
2 \times 2 = 4
\][/tex]
[tex]\[
25 - 1 = 24
\][/tex]
[tex]\[
24 \times 6 = 144
\][/tex]
5. Add the results inside the parentheses:
[tex]\[
4 + 144 = 148
\][/tex]
6. Complete the multiplication and division:
[tex]\[
S_{25} = \frac{25}{2} \times 148
\][/tex]
[tex]\[
S_{25} = 25 \times 74
\][/tex]
7. Multiply to find the sum:
[tex]\[
25 \times 74 = 1850
\][/tex]
Therefore, the sum of the first 25 terms of the given arithmetic sequence is [tex]\( 1850 \)[/tex].
