The first term and second term of a geometric progression are 16 and 2
respectively, Find the sum to infinity of the progression
r = 2/16 = 1/8
S = a/(1-r) = 16/(1 - 1/8) = ____
To find the sum to infinity of a geometric progression, we need to determine if the progression is convergent or divergent.
In a geometric progression, if the absolute value of the common ratio (r) is less than 1, then the progression is convergent. If the absolute value of the common ratio is greater than or equal to 1, then the progression is divergent.
Given that the first term (a₁) is 16 and the second term (a₂) is 2, we can calculate the common ratio (r) using the formula:
r = a₂/a₁
r = 2/16
r = 1/8
Since the absolute value of the common ratio (1/8) is less than 1, the progression is convergent.
Now, to find the sum to infinity (S) of a convergent geometric progression, we can use the formula:
S = a₁ / (1 - r)
S = 16 / (1 - 1/8)
S = 16 / (7/8)
S = 16 * 8/7
S = 128/7
Therefore, the sum to infinity of the given geometric progression is 128/7.
To find the sum to infinity of a geometric progression, we need to determine the common ratio (r) and then use the formula for the sum of an infinite geometric series.
We are given that the first term (a₁) is 16 and the second term (a₂) is 2. We can use these values to find the common ratio (r).
The formula for the common ratio (r) is:
r = a₂ / a₁
Substituting the given values, we have:
r = 2 / 16
Simplifying the fraction:
r = 1 / 8
Now, we can use the formula for the sum to infinity (S) of a geometric progression:
S = a₁ / (1 - r)
Substituting the values, we get:
S = 16 / (1 - 1/8)
Finding the common denominator on the right side:
S = 16 / (8/8 - 1/8)
S = 16 / (7/8)
Dividing the fraction:
S = 16 * 8/7
Calculating the result:
S = 128/7
Therefore, the sum to infinity of the geometric progression is 128/7.