Show work and explain with formulas.

26. Find the sum of the first 6 terms of a geometric series: 80 + (-20) + 5 + ...

27. Find the 4 geometric means between 1/25 and 125.

26)

t2/t1=-20/80=-1/4=-0.25

t3/t2=5 / (-20) = -1/4-0.25

So,

common ratio=-1/4

The formula applied to calculate sum of first n terms of a GP:

Sn=a (rⁿ-1) / (r-1)

S6=80{ (-0.25) ^6-1} / (-0.25-1)

=63.98

27)

1/25,?,?,?,?, 125

where '?' means geometric mean

You can find those missing terms in between 1/25 and 125 to get geometric means.

For that we need to find common ratio by using formula,

tn=t1*r^ (n-1)

here n=last nth term=6, so,

t6 = (1/25) * r^ (6-1)

125 = (1/25) * r^5

125*25=r^5

r=65.6631951101=65.66

thus,

t2=t1*r=65.66/25

t3=t2*r = (65.66/25) * 65.66

=65.66²/25

Similarly,

t4=65.66³/25

t5=65.66⁴/25

t2, t3, t4 and t5 are required geometric means