Find the first 4 terms of the infinite geometric series if S = 33/4 and r = 1/3
The formula for the sum of an infinite geometric sequence is S = a1/(1-r), where a1 is the first term and r is the common ratio.
Since we are given S and r, we can use algebra to solve for a1 as follows:
S = a1/(1-r)
33/4 = a1/(1-1/3)
33/4 = a1/(2/3)
a1 = (33/4) * (2/3)
a1 = 22/4
a1 = 11/2
Therefore, the first term is 11/2 and the common ratio is 1/3. To find the first 4 terms, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1)
So:
a2 = 11/2 * (1/3)^1 = 11/6
a3 = 11/2 * (1/3)^2 = 11/18
a4 = 11/2 * (1/3)^3 = 11/54
Therefore, the first 4 terms are:
11/2, 11/6, 11/18, 11/54.
To find the first 4 terms of the infinite geometric series, we need to know the value of the first term (a) and the common ratio (r). Given that the sum of the series (S) is 33/4 and the common ratio (r) is 1/3, we can find the first term using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Substituting the given values:
33/4 = a / (1 - 1/3)
To simplify the equation, we can combine the fractions on the right side:
33/4 = a / (2/3)
To isolate the variable 'a', we can multiply both sides of the equation by (2/3):
(2/3) * (33/4) = a
Simplifying the equation further:
(2/3) * (33/4) = a
11/2 = a
Now that we know the value of the first term (a = 11/2) and the common ratio (r = 1/3), we can find the first 4 terms of the series:
1st term = a = 11/2
2nd term = a * r = (11/2) * (1/3) = 11/6
3rd term = (11/6) * (1/3) = 11/18
4th term = (11/18) * (1/3) = 11/54
Therefore, the first 4 terms of the series are: 11/2, 11/6, 11/18, and 11/54.
To find the first 4 terms 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, and r is the common ratio.
In this case, we are given that S = 33/4 and r = 1/3. Let's substitute these values into the formula and solve for a:
33/4 = a / (1 - 1/3).
To simplify the equation, we need to find a common denominator:
33/4 = a / (3/3 - 1/3)
33/4 = a / (2/3).
Now, we can solve for a by cross-multiplying:
33/4 = (a * 3) / 2.
Multiply both sides of the equation by 4 and divide by 3 to isolate a:
a = (33/4) * (2/3)
a = 22/4
a = 11/2.
Now that we have the first term, a = 11/2, and the common ratio, r = 1/3, we can find the first 4 terms of the series:
First term: a = 11/2.
Second term: a * r = (11/2) * (1/3) = 11/6.
Third term: (11/2) * (1/3) * (1/3) = 11/18.
Fourth term: (11/2) * (1/3)^3 = 11/54.
Therefore, the first 4 terms of the infinite geometric series are 11/2, 11/6, 11/18, and 11/54.