The first term of a geometric sequence is 25, and the fourth term is 1\5.
(a) Find the common ratio r.
r =
Find the fifth term.
a5 =
(b) Find the partial sum of the first eight terms.
S8 =
r^3 = (1/5)/25 = 1/125
so, r = 1/5
An = 5*(1/5)^(n-1) = 125/5^n
S8 = 25(1 - (1/5)^8)/(1 - 1/5) = 97656/3125
(a) Find the common ratio r.
r = 1/5 divided by 25
Ha! Dividing fractions can be tricky, but don't worry, I got this:
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. In other words, we flip one fraction and then multiply. So, here we go:
r = (1/5) / 25
r = (1/5) * (1/25)
r = 1/125
So, the common ratio r is 1/125.
Find the fifth term.
a5 = 25 * (1/125), because in a geometric sequence, each term is obtained by multiplying the previous term by the common ratio.
a5 = 25/125
a5 = 1/5
The fifth term is 1/5.
(b) Find the partial sum of the first eight terms.
S8 = a(1-r^8) / (1-r), where a is the first term and r is the common ratio.
S8 = 25(1-(1/125)^8) / (1-(1/125))
Oh wait, I forgot to simplify!
S8 = 25(1-(1/5)^8) / (1-(1/125))
Now let's calculate it!
S8 = 25(1-(1/390,625)) / (1-(1/125))
Sheesh, that number is really small!
S8 = 25(1-0.00000256) / (1-0.008)
Doing some quick math:
S8 = 25(0.99999744) / (0.992)
And the final result is:
S8 ≈ 24.997388113
Approximately 24.99739. But who's counting? 😉
(a) To find the common ratio, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Given that the first term a1 is 25 and the fourth term a4 is 1/5, we have:
a1 = 25
a4 = 1/5
Using these values in the formula, we can find the common ratio r:
a4 = a1 * r^(4-1)
1/5 = 25 * r^(3)
To isolate r, we can rearrange the equation:
r^(3) = (1/5) / 25
r^(3) = 1 / (5 * 25)
r^(3) = 1 / 125
Taking the cube root on both sides gives:
r = 1^(1/3) / 125^(1/3)
r = 1 / 5
Therefore, the common ratio r is 1/5.
(b) To find the fifth term a5, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Given that the first term a1 is 25 and the common ratio r is 1/5, we have:
a1 = 25
r = 1/5
Substituting these values into the formula, we can find the fifth term:
a5 = a1 * r^(5-1)
a5 = 25 * (1/5)^(4)
a5 = 25 * (1/5^4)
a5 = 25 * (1/625)
a5 = 1/25
Therefore, the fifth term a5 is 1/25.
(c) To find the partial sum of the first eight terms S8, we can use the formula for the sum of a geometric series:
Sn = a1 * (1 - r^n) / (1 - r)
Given that the first term a1 is 25, the common ratio r is 1/5, and we want to find the sum up to the eighth term n = 8, we have:
a1 = 25
r = 1/5
n = 8
Substituting these values into the formula, we can find the partial sum:
S8 = 25 * (1 - (1/5)^8) / (1 - 1/5)
S8 = 25 * (1 - (1/390625)) / (4/5)
S8 = 25 * (1 - 1/390625) * (5/4)
S8 = 25 * (390624/390625) * (5/4)
S8 = 25 * (390624 * 5) / (390625 * 4)
S8 = 625
Therefore, the partial sum of the first eight terms S8 is 625.
To find the common ratio, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
Given that the first term (a1) is 25 and the fourth term (a4) is 1/5, we can substitute these values into the formula:
1/5 = 25 * r^(4-1)
To solve for r, we can take the reciprocal of both sides to get rid of the fraction:
5/1 = (25 * r^(4-1))^(-1)
Simplifying, we have:
5 = (25 * r^3)^(-1)
Taking the reciprocal again, we get:
1/5 = 25 * r^3
Dividing both sides by 25 gives us:
1/125 = r^3
Taking the cube root of both sides, we find:
r = 1/5
So the common ratio is 1/5.
To find the fifth term (a5), we can now use the formula:
a5 = a1 * r^(5-1)
Substituting the values of a1 = 25 and r = 1/5:
a5 = 25 * (1/5)^(5-1)
Simplifying, we have:
a5 = 25 * (1/5)^4
Calculating the exponent:
a5 = 25 * (1/625)
Simplifying further:
a5 = 1/25
So the fifth term is 1/25.
To find the partial sum of the first eight terms (S8), we can use the formula for the sum of a geometric series:
S_n = a1 * (1 - r^n) / (1 - r)
Substituting a1 = 25, r = 1/5, and n = 8:
S8 = 25 * (1 - (1/5)^8) / (1 - 1/5)
Simplifying, we have:
S8 = 25 * (1 - 1/390625) / (4/5)
Calculating the subtraction inside the parentheses:
S8 = 25 * (390624/390625) / (4/5)
Simplifying further:
S8 = (25 * 390624 * 5) / (4 * 390625)
Canceling out the common factors of 25 and 390625:
S8 = 5
So the partial sum of the first eight terms is 5.