Answer:
[tex]a_{16}=26.5[/tex]
Step-by-step explanation:
Arithmetic series: [tex]a_n=a+(n-1)d[/tex]
Sum of arithmetic series: [tex]S_n=\dfrac{n}{2}[2a+(n-1)d][/tex]
(where a is the first term and d is the common difference)
Given:
[tex]\implies a_{25}=44.5[/tex]
[tex]\implies a+(25-1)d=44.5[/tex]
[tex]\implies a+24d=44.5[/tex]
[tex]\implies S_{30}=765[/tex]
[tex]\implies \dfrac{30}{2}[2a+(30-1)d]=765[/tex]
[tex]\implies 30a+435d=765[/tex]
Rewrite [tex]a+24d=44.5[/tex] to make a the subject:
[tex]\implies a=44.5-24d[/tex]
Substitute found expression for a into [tex]30a+435d=765[/tex] and solve for d:
[tex]\implies 30(44.5-24d)+435d=765[/tex]
[tex]\implies 1335-285d=765[/tex]
[tex]\implies 285d=570[/tex]
[tex]\implies d=2[/tex]
Substitute found value of d into [tex]a=44.5-24d[/tex] and solve for a:
[tex]\implies a=44.5-24(2)=-3.5[/tex]
Therefore:
[tex]\implies a_{16}=-3.5+(16-1)2=26.5[/tex]
