Answer:
Ths sum of the first 38 terms is [tex]\frac{1,045}{2}=522.5[/tex]
Step-by-step explanation:
Arithmetic Sequences
They are identified because any term n is obtained by adding or subtracting a fixed number to the previous term. That number is called the common difference.
The equation to calculate the nth term of an arithmetic sequence is:
[tex]a_n=a_1+(n-1)r[/tex]
Where
an = nth term
a1 = first term
r = common difference
n = number of the term
The sum of n terms of the sequence is
[tex]S_n=\frac{n}{2}( a_1 + a_n)[/tex]
The common difference is found by subtracting two consecutive terms:
[tex]r=a_{n+1}-a_n[/tex]
Using the first two terms:
[tex]r=a_2-a_1=5-\frac{9}{2}=\frac{1}{2}[/tex]
Now we find the term n=38
[tex]\displaystyle a_{38}=\frac{9}{2}+(38-1)\frac{1}{2}[/tex]
[tex]\displaystyle a_{38}=\frac{9}{2}+\frac{37}{2}=\frac{46}{2}=23[/tex]
The sum is:
[tex]\displaystyle S_{38}=\frac{38}{2}( \frac{9}{2} + 23)[/tex]
[tex]\displaystyle S_{38}=19( \frac{9+46}{2})[/tex]
[tex]\displaystyle S_{38}=19( \frac{55}{2})= \frac{1,045}{2}=522.5[/tex]
Ths sum of the first 38 terms is [tex]\mathbf{\frac{1,045}{2}=522.5}[/tex]
