Summing a sequence

1.
S(n) of the 1st sequence : S(n) of the 2nd sequence = (2 + 3n) : (3 + 2n)
Hence,
Let S(n) of the 1st sequence = (2 + 3n)k
and S(n) of the 2nd sequence = (3 + 2n)k

For the 1st sequence:
T(7)
= S(7) - S(6)
= (2 + 3(7))k - (2 + 3(6))k
= 23k - 20k
= 3k

For the 2nd sequence:
T(7)
= S(7) - S(6)
= (3 + 2(7))k - (3 + 2(6))k
= 17k - 15k
= 2k

T(7) of the 1st sequence : T(7) of the 2nd sequence
= 3k : 2k
= 3 : 2

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2.
x-1, 1, x, x2, ......, xn ..... is a geometric sequence.
a = x-1
r = x where -1<x<1

Hence, x-1 + 1 + x + x2 + .... +xn +......
is the sum of geometric sequence to infinitive term with |r| < 1.
x-1 + 1 + x + x2 + .... +xn +......
= a/(1 - r)
= x-1/(1 - x)
= 1/[x(1 - x)]

x-1 + x + x2 + .... +xn +...... = 3.5
(x-1 + 1 + x + x2 + .... +xn +......) - 1 = 3.5
(x-1 + 1 + x + x2 + .... +xn +......) = 4.5
1/[x(1 - x)] = 4.5
1/[x(1 - x)] = 9/2
9x(1 - x) = 2
9x - 9x2 = 2
9x2 - 9x + 2 = 0
(3x - 1)(3x - 2) = 0
3x - 1 = 0 oooroo 3x - 2 = 0
x = 1/3 oooroo x = 2/3
=