Find the sum of the first 8 terms of the following sequence. Round to the nearest hundredth if necessary 6,−4, 8/3

Step-by-step 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

c. prove the lengths are the same.

step-by-step explanation:

you need to prove the segments are congruent. congruent segments have the same length. find the lengths of the segments using the distance formula, and compare them.

answer: c. prove the lengths are the same.

the ratio of white tiles to blue tiles is 48: 16. the ratio in it's simplest form is 3: 1.

step-by-step explanation:

ratio: take the first number and add a colon next to it, then add the second number after that. make sure the ratio order matches up with the written problem!

simplest form: we can simplify the ratio 48 : 16 by dividing both terms by the greatest common factor (gcf).

the gcf of 48 and 16 is 16.

divide both terms by 16:

48 ÷ 16 = 3

16 ÷ 16 = 1

reducing to simplest form:

48 : 16 = 3 : 1