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.
Start by using the geometric formula with r=1/2. Sub in 1/80 as the term value and n=5, then solve for "a" the first term.
Once you have the first term, sub it and the sum of 127/320 into the sum formula and solve for n : )
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To find the number of terms in the geometric progression, we first need to determine the first term and the common ratio. Once we have those values, we can use the formulas for the nth term and the sum of a geometric progression to solve for the number of terms.
Let's start by finding the first term, denoted by 'a', and the common ratio, denoted by 'r'.
From the given information, we know that the common ratio (r) is 1/2, and the fifth term (T₅) is 1/80.
The formula for the nth term of a geometric progression is given by:
Tₙ = a * r^(n-1)
Substituting the given values, we can write the equation as:
1/80 = a * (1/2)^(5-1)
Simplifying this equation, we get:
1/80 = a * (1/2)^4
1/80 = a * (1/16)
1/80 = a/16
Multiplying both sides of the equation by 16, we have:
a = 1/80 * 16
a = 1/80 * 2/2 * 8
a = 1/10
So, the first term (a) is 1/10.
Next, let's use the formula for the sum of a geometric progression to find the number of terms (n).
The formula for the sum of a geometric progression is given by:
Sₙ = a * (1 - r^n) / (1 - r)
Substituting the given value for the sum of all terms (Sₙ) as 127/320, the first term (a) as 1/10, and the common ratio (r) as 1/2, we can write the equation as:
127/320 = (1/10) * (1 - (1/2)^n) / (1 - 1/2)
Simplifying this equation further, we have:
127/320 = (1/10) * (1 - (1/2)^n) / (1/2)
127/320 = (1/10) * (1 - (1/2)^n) * (2/1)
127/320 = (1/5) * (1 - (1/2)^n) * 2
Multiplying both sides of the equation by 320, we get:
127 = (1/5) * (1 - (1/2)^n) * 2 * 320
127 = (1/5) * (1 - (1/2)^n) * 640
127 = (1 - (1/2)^n) * 128
Dividing both sides of the equation by 128, we have:
127/128 = 1 - (1/2)^n
Subtracting 1 from both sides of the equation, we get:
-1/128 = - (1/2)^n
Taking the reciprocals of both sides of the equation, we have:
-128 = 2^n
Taking the logarithm base 2 of both sides of the equation, we get:
log₂(-128) = log₂(2^n)
log₂(-128) = n
Since the logarithm of a negative number is not defined, it means that there is no solution for n in this case. Therefore, there is no possible number of terms in the geometric progression that satisfies all the given conditions.
Hope this explanation helps! Let me know if you have any further questions.