The common ratio of a geometric series
is 1/3
and the sum of the first 5 terms is 121.
a) What is the value of the first term?
b) Write the first 5 terms of the series.
S=t*(1 - r^n)/(1 - r)
121=t*(1 - 1/3^5)/(1 - 1/3)
t1 = 81
81, 27, 9, 3, 1
sum = 121, yep, sure enough :)
To find the value of the first term, we can use the formula for the sum of the first n terms of a geometric series.
The sum of the first n terms of a geometric series is given by the formula:
Sn = a(1 - r^n) / (1 - r)
where Sn is the sum of the first n terms, a is the first term of the series, r is the common ratio, and n is the number of terms.
We are given that the common ratio, r, is 1/3, and the sum of the first 5 terms, Sn, is 121. We can substitute these values into the formula and solve for the first term, a.
a) To find the value of the first term, we can use the formula for the sum of the first 5 terms:
121 = a(1 - (1/3)^5) / (1 - 1/3)
Simplifying this equation:
121 = a(1 - 1/243) / (2/3)
121 = a * (242/243) * (3/2)
Multiplying both sides by (2/3):
121 * (2/3) = a * (242/243)
242/3 = a * (242/243)
Dividing both sides by (242/243):
a = (242/3) * (243/242)
Simplifying:
a = 243
Therefore, the value of the first term is 243.
b) To write the first 5 terms of the series, we know that the common ratio is 1/3 and the first term is 243.
The first term, a = 243
The common ratio, r = 1/3
The formula to find the nth term of a geometric series is given by:
an = a * r^(n-1)
We can substitute the values into this formula to find the first 5 terms:
a1 = 243 * (1/3)^0 = 243 * 1 = 243
a2 = 243 * (1/3)^1 = 243 * 1/3 = 81
a3 = 243 * (1/3)^2 = 243 * 1/9 = 27
a4 = 243 * (1/3)^3 = 243 * 1/27 = 9
a5 = 243 * (1/3)^4 = 243 * 1/81 = 3
Therefore, the first 5 terms of the series are 243, 81, 27, 9, and 3.
To find the value of the first term and write the first 5 terms of the geometric series, we can use the formula for the sum of a finite geometric series:
S = a(1 - r^n) / (1 - r)
Where:
S is the sum of the first n terms,
a is the first term of the series,
r is the common ratio, and
n is the number of terms.
a) To find the value of the first term, we need to use the given information that the common ratio (r) is 1/3, and the sum of the first 5 terms (S) is 121. We can substitute these values into the formula and solve for a.
121 = a(1 - (1/3)^5) / (1 - 1/3)
To simplify, let's evaluate the powers and perform the calculations:
121 = a(1 - 1/243) / (2/3)
121 = a(242/243) / (2/3)
121 = (a * 242 * 3) / (243 * 2)
121 = (a * 726) / 486
121 * 486 = a * 726
58706 = a * 726
Now, we can solve for 'a' by dividing both sides by 726:
a = 58706 / 726
a ≈ 80.814
Therefore, the value of the first term (a) is approximately 80.814.
b) Now that we know the value of the first term (a), we can use the common ratio (1/3) to write the first 5 terms of the series.
The first term (a) is approximately 80.814.
The second term = first term * common ratio = 80.814 * 1/3 = 26.938
The third term = second term * common ratio = 26.938 * 1/3 = 8.979
The fourth term = third term * common ratio = 8.979 * 1/3 = 2.993
The fifth term = fourth term * common ratio = 2.993 * 1/3 = 0.998
Therefore, the first five terms of the geometric series are approximately:
80.814, 26.938, 8.979, 2.993, 0.998