Explanation:
with the introduction of fractions it is clear we are multiplying things to get the sequence.
that makes this a geometric sequence, meaning that every new term is created by multiplying the previous term by a certain constant number, the ratio r.
what do we have to multiply with the first term (6) to get -4 ?
6×r = -4
r = -4/6 = -2/3
to check we use the same factor with the second term and see, if we get the predefined third term :
-4 × -2/3 = 8/3
correct, perfect.
so,
a0 = 6
an = an-1 × -2/3 = a0 × (-2/3)^(n-1) = 6 × (-2/3)^(n-1)
for the sum of a finite number of terms of a geometric sequence the formula is :
Sn = a0×(r^n - 1) / (r - 1) if |r| > 1
Sn = a0×(1 - r^n) / (1 - r) if |r| < 1
Sn = a0×n if r = 1
in our case : |-2/3| = 2/3 < 1
S8 = 6×(1 - (-2/3)⁸) / (1 - (-2/3)) =
= 6×(1 - 256/6561) / (3/3 - -2/3) =
= 6×(6561/6561 - 256/6561) / (5/3) =
= 6×(6305/6561) × (3/5) = 2×(6305/2187) × (3/5) =
= 2×(1261/2187) × 3 = 2×(1261/729) =
= 2522/729 = 3.459533608...
≈ 3.46