5. In a geometric sequence, the sum of the first five terms is 44 and the sum of the next five terms is -11/8. Find the common ratio and first term of the series.
S5 = a(1-r^5)/(1-r) = 44
S10 = a(1-r^10)/(1-r) = 44 - 11/8
now divide
(1-r^10)/(1-r^5) = (44 - 11/8)/44
If that looks tough, note that the numerator is a difference of squares.
a1 + a2 + a3 + a4 + a5 = 44
The sum of a certain number of terms of a geometric sequence:
Sn = a1 * ( 1 - r ^ n ) / ( 1 - r )
In this case you have 5 terms:
a1 + a2 + a3 + a4 + a5 = 44
S5 = a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
The sum of the next five terms is -11/8.
This mean:
a6 + a7 + a8 + a9 + a10 = - 11 / 8
Considering:
a1 + a2 + a3 + a4 + a5 = 44 = 44
and
a6 + a7 + a8 + a9 + a10 = - 11 / 8
You can write:
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 44 + ( - 11 / 8 )
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 352 / 8 - 11 / 8
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 = 341 / 8
This is the sum of 10 terms of a geometric sequence.
You know:
Sn = a1 * ( 1 - r ^ n ) / ( 1 - r )
so:
S10 = a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8
a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8
Now you must solve system of 2 equations with 2 unknow:
a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
a1 * ( 1 - r ^ 10 ) / ( 1 - r ) = 341 / 8
[ a1 / ( 1 - r ) ] * ( 1 - r ^ 5 ) = 44 Divide both sides by ( 1 - r ^ 5 )
a1 / ( 1 - r ) = 44 / ( 1 - r ^ 5 )
[ a1 / ( 1 - r ) ] * ( 1 - r ^ 10 ) = 341 / 8 Divide both sides by ( 1 - r ^ 10 )
a1 / ( 1 - r ) = ( 341 / 8 ) / ( 1 - r ^ 10 )
a1 / ( 1 - r ) = a1 / ( 1 - r )
44 / ( 1 - r ^ 5 ) = ( 341 / 8 ) / ( 1 - r ^ 10 ) Take the reciprocal of both sides
( 1 - r ^ 5 ) / 44 = ( 1 - r ^ 10 ) / ( 341 / 8 )
( 1 - r ^ 5 ) / 44 = 8 * ( 1 - r ^ 10 ) / 341
1 / 44 - r ^ 5 / 44 = ( 8 / 341 )* 1 - ( 8 / 341 ) * r ^ 10
1 / 44 - r ^ 5 / 44 = 8 / 341 - 8 r ^ 10 / 341 Add r ^ 5 / 44 to both sides
1 / 44 - r ^ 5 / 44 + r ^ 5 / 44 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44
1 / 44 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44 Subtract 8 / 341 to both sides
1 / 44 - 8 / 341 = 8 / 341 - 8 r ^ 10 / 341 + r ^ 5 / 44 - 8 / 341
1 / 44 - 8 / 341 = - 8 r ^ 10 / 341 + r ^ 5 / 44
1 * 31 / ( 44 * 31 ) - 8 * 4 / ( 341 * 4 ) = - 8 r ^ 10 * 4 / ( 341 * 4 ) + r ^ 5 * 31 / ( 44 * 31 )
31 / 1364 - 32 / 1364 = - 32 r ^ 10 / 1364 + 31 r ^ 5 / 1364
- 1 / 1364 = - 32 r ^ 10 / 1364 + 31 r ^ 5 / 1364
- 1 / 1364 = ( 1 / 1364 ) * ( - 32 r ^ 10 + 31 r ^ 5 ) Multiply both sides by 1364
- 1 = - 32 r ^ 10 + 31 r ^ 5 Add 1 to both sides by
- 1 + 1 = - 32 r ^ 10 + 31 r ^ 5 + 1
0 = - 32 r ^ 10 + 31 r ^ 5 + 1
- 32 r ^ 10 + 31 r ^ 5 + 1 = 0 Multiply both sides by - 1
32 r ^ 10 - 31 r ^ 5 - 1 = 0
32 r ^ 5 * r ^ 5 - 31 r ^ 5 - 1 = 0
32 ( r ^ 5 ) ^ 2 - 31 r ^ 5 - 1 = 0
Substitute r ^ 5 = x
32 x ^ 2 - 31 x - 1 = 0
The solutions are :
x = - 1 / 32
and
x = 1
Now:
For x = - 1 / 32
r ^ 5 = x
r = fifth root ( x ) = fifth root ( - 1 / 32 ) = - 1 / 2
and
For x = 1
r ^ 5 = x
r = fifth root ( x ) = fifth root ( 1 ) = 1
The solutions are:
r = - 1 / 2 and r = 1
Solution r = 1 you must discard becouse for r = 1 you get:
a2 = a1 * r = a1 * 1 = a1
a3 = a2 * r = a1 * 1 = a1
a4 = a3 * r = a1 * 1 = a1 etc.
For r = 1 geometric sequence is:
a1, a1, a1, a1...
This is a constant sequence and you must discard this sequence.
So your solution is: r = - 1 / 2
Replace this value in equation:
S5 = a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
Since the ( - 1 / 2 ) ^ 5 = - 1 / 32
you get:
a1 * ( 1 - r ^ 5 ) / ( 1 - r ) = 44
a1 * ( 1 - ( - 1 / 32 ) ) / ( 1 - ( - 1 / 2 ) ) = 44
a1 * ( 1 + 1 / 32 ) / ( 1 + 1 / 2 ) = 44
a1 * ( 32 / 32 + 1 / 32 ) / ( 2 / 2 + 1 / 2 ) = 44
a1 * ( 33 / 32 ) / ( 3 / 2 ) = 44 Multiply both sides by ( 3 / 2 )
a1 * ( 3 / 2 ) * ( 33 / 32 ) / ( 3 / 2 ) = 44 * ( 3 / 2 )
a1 * 33 / 32 = 132 / 2
33 a1 / 32 = 132 / 2
33 a1 / 32 = 66 Multiply both sides by 32
33 a1 * 32 / 32 = 66 * 32
33 a1 = 2112 Divide both sides by 33
a1 = 2112 / 33
a1 = 64
Your geometric sequence:
64, 64 * ( - 1 / 2 ), 64 * ( - 1 / 2 ) ^ 2, 64 * ( - 1 / 2 ) ^ 3, 64 * ( - 1 / 2 ) ^ 4, 64 * ( - 1 / 2 ) ^ 5, 64 * ( - 1 / 2 ) ^ 6, 64 * ( - 1 / 2 ) ^ 7, 64 * ( - 1 / 2 ) ^ 8, 64 * ( - 1 / 2 ) ^ 9
64, - 32, 16, - 8, 4, - 2, 1, - 1 / 2, 1 / 4, - 1 / 8
Proof:
a1 + a2 + a3 + a4 + a5 =
64 + ( - 32 ) + 16 + ( - 8 ) + 4 =
64 - 32 + 16 - 8 + 4 = 44
a6 + a7 + a8 + a9 + a10 = - 11 / 8
- 2 + 1 + ( - 1 / 2 ) + 1 / 4 + ( - 1 / 8 ) =
- 2 + 1 - 1 / 2 + 1 / 4 - 1 / 8 =
- 2 * 8 / 8 + 1 * 8 / 8 - 1 * 4 / ( 2 * 4 ) + 1 * 2 / ( 4 * 2 ) - 1 / 8 =
- 16 / 8 + 8 / 8 - 4 / 8 + 2 / 8 - 1 / 8 = - 11 / 8
solve for the sum of the geometric sequence of the given: a1 = 2, r = 5, find S10
Why did the geometric sequence bring a calculator to the party?
Because it wanted to sum things up!
Now, let's get down to business. We can use the formulas for the sum of a geometric series to solve this problem. The sum of the first five terms can be expressed as:
S5 = a(1 - r^5) / (1 - r)
And the sum of the next five terms can be expressed as:
S10 = a(1 - r^10) / (1 - r)
Given that S5 = 44 and S10 = -11/8, we can set up the following equations:
44 = a(1 - r^5) / (1 - r)
-11/8 = a(1 - r^10) / (1 - r)
Now, let's get serious for a moment and solve this system of equations to find the values of the common ratio (r) and the first term (a).
To solve this problem, we can use the formula for the sum of terms in a geometric sequence.
The sum of the first n terms of a geometric sequence is given by the formula:
Sn = a * (1 - r^n) / (1 - r)
Where Sn represents the sum of the first n terms, a is the first term of the sequence, and r is the common ratio.
Given that the sum of the first five terms is 44, we can substitute these values into the formula:
44 = a * (1 - r^5) / (1 - r) - Equation 1
Similarly, given that the sum of the next five terms is -11/8, we can write another equation:
-11/8 = a * (1 - r^10) / (1 - r) - Equation 2
Now, we have a system of two equations with two unknowns (a and r). We can solve this system to find the values of a and r.
Let's solve the system of equations step by step:
Step 1: Multiply both sides of Equation 2 by (-8) to eliminate fractions:
11/8 = a * (1 - r^10) / (r - 1)
-8 * (11/8) = a * (1 - r^10) / (r - 1)
-11 = a * (1 - r^10) / (r - 1) - Equation 3
Step 2: Multiply both sides of Equation 1 by (-11) to simplify the terms:
-11 * 44 = a * (1 - r^5) / (r - 1)
-484 = a * (1 - r^5) / (r - 1) - Equation 4
Step 3: Divide Equation 3 by Equation 4 to eliminate 'a':
(-11) / (-484) = (1 - r^10) / (1 - r^5)
11/484 = (1 - r^10) / (1 - r^5)
Now, we have a new equation in terms of 'r' only. We can solve this equation to find the value of 'r'.
To solve this equation, let's multiply both sides by (1 - r^5):
(11/484) * (1 - r^5) = 1 - r^10
Multiplying and rearranging terms further:
11 - 11 * r^5 / 484 = 1 - r^10
11 * r^5 / 484 - r^10 = -10
To continue solving for 'r', we can make a substitution:
Let x = r^5
The equation now becomes:
11 * x / 484 - x^2 = -10
Rearranging terms further:
x^2 - (11/484) * x - 10 = 0
Now, we have a quadratic equation. We can solve it using factoring, completing the square, or the quadratic formula. Once we solve for 'x', we can substitute it back to find 'r'. Finally, we can substitute 'r' into Equation 1 or Equation 2 to find 'a'.