Question

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.

Answer 1

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.

Answer 2

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

Answer 3

solve for the sum of the geometric sequence of the given: a1 = 2, r = 5, find S10