To find the indicated sum [tex]\( S_4 \)[/tex] for the sequence where the general term is given by [tex]\( t_n = 2n - 3 \)[/tex], follow these steps:
1. Identify the general term: The term [tex]\( t_n \)[/tex] of the sequence is given by the formula [tex]\( t_n = 2n - 3 \)[/tex].
2. Calculate the first four terms: Compute [tex]\( t_1, t_2, t_3, \)[/tex] and [tex]\( t_4 \)[/tex] using the general term formula.
- For [tex]\( n = 1 \)[/tex]:
[tex]\[
t_1 = 2(1) - 3 = 2 - 3 = -1
\][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[
t_2 = 2(2) - 3 = 4 - 3 = 1
\][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[
t_3 = 2(3) - 3 = 6 - 3 = 3
\][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
t_4 = 2(4) - 3 = 8 - 3 = 5
\][/tex]
3. Sum the first four terms: Now, sum up the values of [tex]\( t_1, t_2, t_3, \)[/tex] and [tex]\( t_4 \)[/tex].
[tex]\[
S_4 = t_1 + t_2 + t_3 + t_4
\][/tex]
Substituting the values we computed:
[tex]\[
S_4 = (-1) + 1 + 3 + 5
\][/tex]
4. Calculate the sum: Perform the addition step-by-step.
- Sum of the first two terms:
[tex]\[
(-1) + 1 = 0
\][/tex]
- Sum of the next term with the current sum:
[tex]\[
0 + 3 = 3
\][/tex]
- Sum of the next term with the current sum:
[tex]\[
3 + 5 = 8
\][/tex]
Therefore, the indicated sum [tex]\( S_4 \)[/tex] is [tex]\( 8 \)[/tex].
