The first and second terms of a geometric sequence are respectively the first and third terms of a linear sequence The fourth term of the linear sequence is 10 and the sum of its first five term is 60 Find the first five terms of the linear sequence?

A typical geometric sequence can be represented as

#c_0a, c_0a^2,cdots,c_0a^k#

and a typical arithmetic sequence as

#c_0a, c_0a+Delta, c_0a+2Delta,cdots,c_0a+kDelta#

Calling #c_0 a# as the first element for the geometric sequence we have

#{ (c_0 a^2= c_0a+2Delta -> "First and second of GS are the first and third of a LS"), (c_0a+3Delta=10->"The fourth term of the linear sequence is 10"), (5c_0a+10Delta=60->"The sum of its first five term is 60"):}#

Solving for #c_0,a,Delta# we obtain

#c_0=64/3, a=3/4, Delta=-2# and the first five elements for the arithmetic sequence are

#{16, 14, 12, 10, 8}#

(Ignoring the geometric sequence)

If the linear series is denoted as #a_i : a_1, a_2, a_3, ...#
and the common difference between terms is denoted as #d#
then
note that #a_i=a_1+(i-1)d#

Given fourth term of linear series is 10
#rarr color(white)("xxx") a_1+3d=10color(white)("xxx")[1]#

Given sum of the first 5 terms of the linear sequence is 60
#sum_(i=1)^5 a_i ={:(color(white)(+)a_1),(+a_1+d),(+a_1+2d),(+a_1+3d),(ul(+a_1+4d)),(5a_1+10d):}=60color(white)("xxxx")[2]#

Multiplying [1] by 5
#5a_1+15d=50color(white)("xxxx")[3]#
then subtracting [3] from [2]
#color(white)(-"(")5a_1+10d=60#
#ul(-" ("5a_1+15d=50")")#
#color(white)("xxXXXxx")-5d=10color(white)("xxx")rarrcolor(white)("xxx")d=-2#

Substituting #(-2)# for #d# in [1]
#a_1+3xx(-2)=10color(white)("xxx")rarrcolor(white)("xxx")a_1=16#

From there it follows that the first 5 terms are:
#color(white)("XXX")16, 14, 12, 10, 8#