To find the sum of the first 20 terms of the sequence [tex]\( a_n = 7n + 2 \)[/tex], follow these steps:
1. Identify the first term, [tex]\( a_1 \)[/tex]:
Substitute [tex]\( n = 1 \)[/tex] into the formula for the sequence.
[tex]\[
a_1 = 7(1) + 2 = 7 + 2 = 9
\][/tex]
2. Identify the 20th term, [tex]\( a_{20} \)[/tex]:
Substitute [tex]\( n = 20 \)[/tex] into the formula for the sequence.
[tex]\[
a_{20} = 7(20) + 2 = 140 + 2 = 142
\][/tex]
3. Use the formula for the sum 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 given by:
[tex]\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\][/tex]
Substitute [tex]\( n = 20 \)[/tex], [tex]\( a_1 = 9 \)[/tex], and [tex]\( a_{20} = 142 \)[/tex] into the formula.
[tex]\[
S_{20} = \frac{20}{2} \times (9 + 142) = 10 \times 151 = 1510
\][/tex]
Therefore, the sum of the first 20 terms of the sequence [tex]\( a_n = 7n + 2 \)[/tex] is [tex]\( 1510 \)[/tex].
