# What does "a third" usually means?

Marco went on a hiking trip. The first day he walked 24 kilometers. Each day since, he walked a third of what he walked the day before.

Which expression gives the total distance Marco walked in the first n days of his trip?

This is a geometric sequence problem

given a = 24

What would be my common ratio? I don't understand what does "a third" mean on the statement.  does it mean 1/3? or what?

Relevance

''a third'' means and has always meant, (1/3).

Suppose we have a GP (geometric progression) of form : a,ar,ar^2,....,ar^(n-1),....

This can also be written as t(1),t(2),t(3),....,t(n),.. where t(1) = a, t(n) = ar^(n-1), and

r is the common quotient between sequential terms, ie., r = t(n+1)/t(n), n = 1,2,3,...

Sum of the 1st n terms of a GP is S(n) = a(1-r^n)/(1-r). Here, a = 24 and r = (1/3).

Therefore S(n) = {24[1 -(1/3)^n]/[1 - (1/3)]}km  = 36[1 - (1/3)^n]km.

• Anonymous
4 weeks ago

Third means 2 things.

Cardinal and ordinal.

• it means a third you idiot , go back to school

•  Marco went on a hiking trip.

The first day he walked 24 kilometers.

Each day since then,

he walked a third of what he walked the day before.

The expression which gives the total distance Marco walked

in the first n days of his trip:

given a = 24,  the common ratio is 1/3

24, 8, 8/3, 8/9, ...

a_n = 8 3^(2 - n) (for all terms given)

• One part out of three parts.

• ''third of'' means x by 1/3

so, we have: 24, 8, 8/3, 8/9,....

Now, Sₙ = a(1 - rⁿ)/(1 - r)...where a is the first term and r is the common ratio.

so, with a = 24 and r = 1/3 we have:

Sₙ = 24(1 - rⁿ)/(2/3) = 36(1 - rⁿ)

Also, S∞ = a/(1 - r)

Then, S∞ = 24/(2/3) = 36

This means if he continues he will never actually walk a total of 36 km...but close!!

:)>

• 1 third is 1 divided by 3 or 1/3 or 0.333...

total distance = 24 + (1/3)24 + (1/3)(1/3)24 + (1/3)(1/3)(1/3)24 ...

or 24 + 24/3¹ + 24/3² + 24/3³ + ...

or 24 + 8 + 2.66... + 0.648... + ...

S = a(1 – rⁿ)/(1 – r)

S = 24(1 – (1/3)ⁿ)/(1 – (1/3))

S = 24(1 – (1/3)ⁿ)/(2/3)

S = 36(1 – (1/3)ⁿ)

test

n=1, S = 36 – 12 = 24

n=2, S = 36 – 4 = 32

n=3, S = 36 – 1.33 = 34.666..

ok

Geometric Series is one where the ratio of successive terms in the

series is constant (r). a is the first term.

a + ar + ar² + ar³ + ar⁴ + ...

Sum of first n terms is

S = a(1 – rⁿ)/(1 – r)

• Anonymous
4 weeks ago

Divide your total by three and you have your answer. Example day one he travels 24km, day two he travelled on third (simply divide 24 by 3 and you have the answer) for day three you divide day two result by 3. Then you add up the totals for each day and that's your answer

• He walked 24 km the first day

The next day, he walked 8 km  (because 1/3 of 24 is 8)

Then he walked 8/3 km

Then he walked 8/9 km

And so on

S = 24 + (24/3) + (24/9) + (24/27) + ....

S = 24 + (1/3) * (24 + 24/3 + 24/9 + 24/27 + ...)

S = 24 + (1/3) * S

S - (1/3) * S = 24

(2/3) * S = 24

S = 24 * (3/2)

S = 36

If he walked for an infinite number of days, he'd walk 36 km

S = 24 * (1/3)^(1 - 1) + 24 * (1/3)^(2 - 1) + 24 * (1/3)^(3 - 1) + ... + 24 * (1/3)^(n - 1)

S = 72 * (1/3) + 72 * (1/3)^2 + ... + 72 * (1/3)^n

S = 72 * (1/3 + (1/3)^2 + ... + (1/3)^n)

T = (1/3) + (1/3)^2 + ... + (1/3)^n

T * (1/3) = (1/3)^2 + (1/3)^3 + ... + (1/3)^n + (1/3)^(n + 1)

T - T * (1/3) = (1/3) + (1/3)^2 - (1/3)^2 + ... + (1/3)^n - (1/3)^n - (1/3)^(n + 1)

T * (1 - 1/3) = (1/3) - (1/3) * (1/3)^n

T * (2/3) = (1/3) * (1 - (1/3)^n)

T = (1/2) * (1 - (1/3)^n)

S = 72 * T

S = 72 * (1/2) * (1 - (1/3)^n)

S = 36 * (1 - (1/3)^n)

You'll walk 36 * (1 - (1/3)^n) km after n number of days.

1mm = 10^(-3) m = 10^(-6) km

36 - 10^(-6) = 36 * (1 - (1/3)^n) =>

36 - 10^(-6) = 36 - 36 * (1/3)^n

36 * (1/3)^n = 10^(-6)

36 * 10^6 = 3^n

36,000,000 = 3^n

ln(36,000,000) = n * ln(3)

n = ln(36,000,000) / ln(3)

n = 15.837279152879222499978459360814

In 16 days, you're going to be within 1mm of the 36km total.  It takes the remainder of eternity to make up that last millimeter.  Geometric sequences are crazy.

• S= 24[1 - (1/3)^n] / (2/3)