Question

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?

Answer 1

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

Explanation:

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}`

Answer 2

first 5 terms of the linear sequence: `color(red)({16,14,12,10,8})`

Explanation:

(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`