Determine the sum of the infinite geometric series 28 + 14/3 + 7/9 + 7/54 ?
clearly,
a = 28
r = 1/6
So, S = a/(1-r) = 28/(5/6) = 33.6
To find the sum of an infinite geometric series, we use the formula:
S = a / (1 - r),
where S is the sum, a is the first term, and r is the common ratio.
In this case, the first term (a) is 28, and the common ratio (r) is (14/3) / 28.
To simplify, we divide both the numerator and denominator of (14/3) by 14:
r = (14/3) / 28
= (2/3) / 2
= 1/3
Now we can substitute the values into the formula to find the sum:
S = a / (1 - r)
= 28 / (1 - 1/3).
To simplify the expression on the denominator, we multiply (1/3) by (3/3) to get a common denominator:
S = 28 / (1 - 1/3)
= 28 / (3/3 - 1/3)
= 28 / (2/3)
= 28 * (3/2)
= 42.
Therefore, the sum of the infinite geometric series 28 + 14/3 + 7/9 + 7/54 is equal to 42.
To determine the sum of an infinite geometric series, we need to use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where:
S is the sum of the series,
a is the first term of the series,
and r is the common ratio.
In this case, the first term (a) is 28, and the common ratio (r) can be found by dividing any term by its preceding term. Let's calculate the common ratio:
r = (14/3) / 28
r = (14/3) * (1/28)
r = 1/6
Now that we have the common ratio, we can calculate the sum of the infinite geometric series. Plugging in the values into the formula, we get:
S = 28 / (1 - 1/6)
To simplify the calculation, we need to find a common denominator for the fraction:
S = 28 / (6/6 - 1/6)
S = 28 / (5/6)
Then, we can multiply the numerator by the reciprocal of the denominator to get the result:
S = 28 * (6/5)
S = 168/5
Therefore, the sum of the given infinite geometric series is 168/5 or 33.6.