Answer :
Answer:
The signa notation to represent the first five ten f(x) is given by [tex] f(x)=\sum\limits^{5}_{n=1}a_n[/tex]
Step-by-step explanation:
Given sequence is -5, -9, 13...
Let f(x) be the given sequence and is denoted by
[tex]f(x)=\{-5,-9,-13,...\}[/tex]
Let the first term be [tex]a_1, 2^{\textrm{nd}}[/tex] term be [tex]a_3,...[/tex]
ie, [tex]a_1=-5,a_2=-9, a_3=-13,...[/tex]
To find the common difference d:
[tex]d=a_2-a_1[/tex]
[tex]=-9-(-5)[/tex]
[tex]=-9-(+5)[/tex]
[tex]d=-4[/tex]
[tex]d=a_3-a_2[/tex]
[tex]=-13-(-9)[/tex]
[tex]=-13+9[/tex]
[tex]d=-4[/tex]
Therefore the common difference d is -4 for given sequence f(x) with [tex]a_1=-5[/tex] and d=-4, the seqence f(x) is an arithmetic sequence
By defintion of arithmetic sequence
[tex]a_n=a+(n-1)d\hfill(1)[/tex]
Now to find [tex]a_4, a_5[/tex]:
put n=4 in equation (1)
[tex]a_4=a+(4-1)d[/tex]
[tex]a_4=-5+(3)(-4)[/tex] [since a=-5, d=-4]
[tex]=-5-12[/tex]
[tex]a_4=-17[/tex]
in equation (1)
[tex]a_5=a+(5-1)d[/tex]
[tex]a_5=-5+(4)(-4)[/tex] [since a=-5, d=-4]
[tex]=-5-16[/tex]
[tex]a_5=-21[/tex]
Therefore [tex]a_4=-17[/tex] and [tex]a_5=-21[/tex]
Therefore [tex]f(x)=\{-5,-9,-13,-17,-21,...\}[/tex]
Now to represent the sum of the first five terms of f(x) using sigma notation as below
[tex]f(x)=\sum_{n=1}^5 a_n[/tex]
where [tex]\sum_{n=1}^5 a_n=a_1+a_2+a_3+a_4+a_5[/tex]
[tex]-5-9-13-17-21[/tex]
[tex]\sum_{n=1}^5 a_n=-65[/tex]