The common ratio of a geometric progression is 1/2 , the fifth term is 1/80 , and the sum of all of its terms is 127/320 . Find the number of terms in the progression.
term5 is 1/80 and r = 1/2
so
a(1/2)^4 = 1/80
a(1/16) = 1/80
a = 1/5
so the terms are: 1/5, 1/10, 1/20, 1/40, 1/80, ....
sum(n) = 127/320
(1/5)((1/2)^n - 1) / (1/2 - 1) = 127/320
(-2/5)((1/2)^n -1) = 127/320
(1/2)^n - 1 = -635/640 = -127/128
(1/2)^n = 1 - 127/128 = 1/128 = (1/2)^7
n = 7 , there were 7 terms
To find the number of terms in the geometric progression, we need to use two pieces of information: the common ratio and the sum of all the terms.
Step 1: Find the first term (a) of the geometric progression.
The fifth term (a₅) is given as 1/80, and the common ratio (r) is given as 1/2.
We can use the formula for the nth term of a geometric progression to find the first term (a):
a₅ = a * r^(n-1)
1/80 = a * (1/2)^(5-1)
1/80 = a * (1/2)^4
1/80 = a * 1/16
a = (1/80) * 16
a = 1/5
Step 2: Find the common ratio (r).
The common ratio (r) is given as 1/2.
Step 3: Find the number of terms (n) in the geometric progression.
The sum of all the terms in the geometric progression is given as 127/320.
We can use the formula for the sum of a geometric series to find the number of terms (n):
S = a * (1 - r^n) / (1 - r)
127/320 = (1/5) * (1 - (1/2)^n) / (1 - (1/2))
127/320 = (1/5) * (1 - (1/2)^n) / (1/2)
(127/320) * (2/1) = (1/5) * (1 - (1/2)^n)
127/160 = (1 - (1/2)^n)
(1/2)^n = 1 - 127/160
(1/2)^n = 33/160
To solve for n, take the logarithm of both sides:
log((1/2)^n) = log(33/160)
n * log(1/2) = log(33/160)
n = log(33/160) / log(1/2)
Using a calculator, we can find that n ≈ 4.05.
Since the number of terms (n) must be a whole number, the closest whole number to 4.05 is 4.
Therefore, the number of terms in the geometric progression is 4.
To find the number of terms in the geometric progression, we need to first find the first term (a) and the common ratio (r).
Given that the common ratio is 1/2, we have r = 1/2.
We can find the first term (a) using the fifth term (T5) and the common ratio (r):
T5 = a * r^4
Substituting the given values, we have:
1/80 = a * (1/2)^4
1/80 = a * (1/16)
1/80 = a/16
a = (1/80) * 16
a = 1/5
Now that we know the first term (a) is 1/5 and the common ratio (r) is 1/2, we can find the number of terms (n) using the sum of all the terms formula.
The sum of all terms (S) in a geometric progression can be calculated using the formula:
S = a * (1 - r^n) / (1 - r)
We are given that the sum of all terms is 127/320. Substituting the known values, we have:
127/320 = (1/5) * (1 - (1/2)^n) / (1 - 1/2)
Multiplying both sides by (1 - 1/2) to eliminate the denominator on the right side, we get:
127/320 = (1/5) * (1 - (1/2)^n) * 2/1
Simplifying further, we have:
127/320 = (2/5) * (1 - (1/2)^n)
Dividing both sides by (2/5), we get:
(127/320) / (2/5) = 1 - (1/2)^n
Simplifying further, we have:
(127/320) * (5/2) = 1 - (1/2)^n
Now we can simplify the left side of the equation:
127/64 = 1 - (1/2)^n
Rearranging the equation, we have:
(1/2)^n = 1 - 127/64
Simplifying further, we have:
(1/2)^n = (64 - 127)/64
(1/2)^n = -63/64
Now we can take the logarithm of both sides to solve for n:
log[(1/2)^n] = log(-63/64)
Using the logarithmic identity, we have:
n * log(1/2) = log(-63/64)
Dividing both sides by log(1/2), we get:
n = log(-63/64) / log(1/2)
At this point, we encounter an issue because the logarithm of a negative number is undefined. Therefore, there is no solution for the number of terms in this geometric progression using the given information.