The sum of the first five terms of a geometric series is 186 and the sum of the first six terms is 378. if the fourth term is 48, determine a(first term),r(ratio), t10, S10.
if sum(6) = 378 and sum(5) = 186
then term(6) = 378-186 = 192
so
ar^5 = 192
ar^3 = 48
divide them
r^2 = 4
r = ±2
if r=2, then a(8) = 48 --->a = 6
if r = -2, then a(-8) = 48 --- a = -6
if a= 6, r=2, t(10) = 6(2^9) = 3072
if a= -6, r=-2 , t(10) = -6(-2)^9 = 3072
if a=6, r=2, sum(10) = 6(2^10 - 1)/1 = 6138
if a=-6,r=-2, sum(1) = -6((-2)^10 - 1)/-2-1) = 2046
Well, well, well, looks like we have some number crunching to do! Let's get started, shall we?
First, we need to find the common ratio (r). To do this, we'll divide the sum of the sixth term by the sum of the fifth term: 378 / 186 = 2.
Now that we know the ratio, we can find the first term (a). We'll use the formula for the sum of a geometric series:
S_n = a(1 - r^n) / (1 - r)
Plugging in the known values, we have:
186 = a(1 - 2^5) / (1 - 2)
Simplifying this equation, we get:
186 = a(-31) / (-1)
Cross-multiplying, we find:
186 = 31a
Dividing both sides by 31, we find:
a = 6
So, the first term (a) is 6.
Now, we move on to finding T10 (the 10th term) and S10 (the sum of the first 10 terms). We'll use the formula for the nth term of a geometric series:
T_n = a * r^(n-1)
For T10, we have:
T10 = 6 * 2^(10-1) = 6 * 2^9 = 6 * 512 = 3072
Therefore, the 10th term (T10) is 3072.
To find S10, we'll use the formula for the sum of a geometric series:
S_n = a(1 - r^n) / (1 - r)
Plugging in our known values, we have:
S10 = 6(1 - 2^10) / (1 - 2)
Simplifying this equation, we get:
S10 = 6(-1023) / (-1)
Dividing both sides, we find:
S10 = 6 * 1023 = 6138
So, the sum of the first 10 terms (S10) is 6138.
There you have it! The first term (a) is 6, the common ratio (r) is 2, the 10th term (T10) is 3072, and the sum of the first 10 terms (S10) is 6138. Voila!
To find the first term (a) and the ratio (r) of a geometric series, we can create a system of equations using the given sums.
Let's assume the first term (a) and the ratio (r) of the geometric series.
The sum of the first five terms can be expressed as:
S5 = a + ar + ar^2 + ar^3 + ar^4
Given that S5 = 186, we have the equation:
186 = a + ar + ar^2 + ar^3 + ar^4 ---(Equation 1)
The sum of the first six terms can be expressed as:
S6 = a + ar + ar^2 + ar^3 + ar^4 + ar^5
Given that S6 = 378, we have the equation:
378 = a + ar + ar^2 + ar^3 + ar^4 + ar^5 ---(Equation 2)
We are also given that the fourth term equals 48:
ar^3 = 48 ---(Equation 3)
Now, let's solve the system of equations:
From Equation 3, we can isolate a:
a = 48/r^3
Substitute this value of a in Equations 1 and 2:
186 = (48/r^3) + (48/r^2) + (48/r) + (48/r^3) + (48/r^4) ---(Equation 1)
378 = (48/r^3) + (48/r^2) + (48/r) + (48/r^3) + (48/r^4) + (48/r^5) ---(Equation 2)
Now, we can solve this system of equations to find the value of r.
To find the values of a (first term), r (common ratio), t10 (tenth term), and S10 (sum of the first ten terms) in the geometric series, we can use the given information and formulas for geometric series.
Let's start by finding the value of r (common ratio):
We know that the sum of the first five terms is 186 and the sum of the first six terms is 378.
Using the formula for the sum of a geometric series, we can set up the following equations:
S5 = a(1 - r^5) / (1 - r) = 186
S6 = a(1 - r^6) / (1 - r) = 378
Dividing these two equations, we get:
S6 / S5 = (a(1 - r^6) / (1 - r)) / (a(1 - r^5) / (1 - r))
378 / 186 = (1 - r^6) / (1 - r^5)
Simplifying the equation:
2.032 = (1 - r^6) / (1 - r^5) (1)
Now, we know that the fourth term is 48. The general formula for the nth term of a geometric series is:
tn = a * r^(n-1)
Substituting n = 4 and tn = 48 into the formula, we have:
48 = a * r^3 (2)
To solve equations (1) and (2), we can use substitution or an algebraic method. Let's use substitution:
From equation (2), we can express a in terms of r:
a = 48 / r^3
Substituting this expression into equation (1):
2.032 = (1 - r^6) / (1 - r^5)
2.032 = (1 - r^6) / (1 - r^5) * (r^3 / 48)
Simplifying:
2.032 = (1 - r^6) / (1 - r^5) * (r^3 / 48)
2.032 = (1 - r^6) * (r^3) / (48 * (1 - r^5))
Cross-multiplying:
2.032 * 48 * (1 - r^5) = (1 - r^6) * (r^3)
Expanding:
97.536 - 98.624 r^5 = r^3 - r^9
Rearranging:
r^9 - r^3 - 98.624 r^5 + 97.536 = 0
We now have a ninth-degree polynomial equation for r. Since it does not have a simple factorization or solution, we need to use numerical methods or calculators to find the approximate value of r.
Once we find the value of r, we can substitute it back into equation (2) to solve for a, and then use the formulas for the nth term (t10) and the sum of the first ten terms (S10) to find their values.
Please note that in this case, finding the exact values of a, r, t10, and S10 may be challenging due to the ninth-degree polynomial equation. Therefore, using numerical methods is the best approach to determine approximate values.