Given the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.

Question

15) a_1 = 0.8, r = −5

16) a_1 = 1, r = 2

Answer 1

The n th term of a geometric sequence :

The n th term of a geometric sequence is  ---------> (1)

Where  is the first term and r  is the common ratio.

1)

Given  =  0.8 and common ratio is r = - 5

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  0.8(-5)^(n-1)

a1  =   0.8(-5)^(1-1)

      =   0.8(-5)^(0)

      =   0.8

a2  =   0.8(-5)^(2-1)

      =   0.8(-5)^(1)

      =   0.8(-5)

      =  - 4

a3  =   0.8(-5)^(3-1)

      =   0.8(-5)^(2)

      =   0.8(25)

      =  20

a4  =   0.8(-5)^(4-1)

      =   0.8(-5)^(3)

      =   0.8(125)

      =  100

a5  =   0.8(-5)^(5-1)

      =   0.8(-5)^(4)

      =   0.8(125)

      =  500

The GP is 0.8, -4, 20, 100, 500

2)

Given  = 1 and common ratio is r = 2

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  1(2)^(n-1)

a_n  =  2^(n-1)

a1  =  2^(1-1)

      =   2^(0)

      =   1

a2  =  2^(2-1)

      =   2^(1)

      =   2

a3  =  2^(3-1)

      =   2^(2)

      =   4

a4  =  2^(4-1)

      =   2^(3)

      =   8

a5  =  2^(5-1)

      =   2^(4)

      =   16

The GP is 1, 2 , 4, 8, 16

Answer :

1)  

The n th term of a geometric sequence is a_n  =  0.8(-5)^(n-1)

The GP is 0.8, -4, 20, 100, 500

2)

The n th term of a geometric sequence is a_n  =  2^(n-1)

The GP is 1, 2 , 4, 8, 16.

Answer 2

The n th term of a geometric sequence :

The n th term of a geometric sequence is  ---------> (1)

Where  is the first term and r  is the common ratio.

1)

Given  =  0.8 and common ratio is r = - 5

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  0.8(-5)^(n-1)

a1  =   0.8(-5)^(1-1)

      =   0.8(-5)^(0)

      =   0.8

a2  =   0.8(-5)^(2-1)

      =   0.8(-5)^(1)

      =   0.8(-5)

      =  - 4

a3  =   0.8(-5)^(3-1)

      =   0.8(-5)^(2)

      =   0.8(25)

      =  20

a4  =   0.8(-5)^(4-1)

      =   0.8(-5)^(3)

      =   0.8(125)

      =  100

a5  =   0.8(-5)^(5-1)

      =   0.8(-5)^(4)

      =   0.8(125)

      =  500

The GP is 0.8, -4, 20, 100, 500

2)

Given  = 1 and common ratio is r = 2

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  1(2)^(n-1)

a_n  =  2^(n-1)

a1  =  2^(1-1)

      =   2^(0)

      =   1

a2  =  2^(2-1)

      =   2^(1)

      =   2

a3  =  2^(3-1)

      =   2^(2)

      =   4

a4  =  2^(4-1)

      =   2^(3)

      =   8

a5  =  2^(5-1)

      =   2^(4)

      =   16

The GP is 1, 2 , 4, 8, 16

Answer :

1)  

The n th term of a geometric sequence is a_n  =  0.8(-5)^(n-1)

The GP is 0.8, -4, 20, 100, 500

2)

The n th term of a geometric sequence is a_n  =  2^(n-1)

The GP is 1, 2 , 4, 8, 16.

Answer 3

The n th term of a geometric sequence :

The n th term of a geometric sequence is  ---------> (1)

Where  is the first term and r  is the common ratio.

1)

Given  =  0.8 and common ratio is r = - 5

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  0.8(-5)^(n-1)

a1  =   0.8(-5)^(1-1)

      =   0.8(-5)^(0)

      =   0.8

a2  =   0.8(-5)^(2-1)

      =   0.8(-5)^(1)

      =   0.8(-5)

      =  - 4

a3  =   0.8(-5)^(3-1)

      =   0.8(-5)^(2)

      =   0.8(25)

      =  20

a4  =   0.8(-5)^(4-1)

      =   0.8(-5)^(3)

      =   0.8(125)

      =  100

a5  =   0.8(-5)^(5-1)

      =   0.8(-5)^(4)

      =   0.8(125)

      =  500

The GP is 0.8, -4, 20, 100, 500

2)

Given  = 1 and common ratio is r = 2

Substitute  =  0.8 and  r = - 5 in equation (1)

a_n  =  1(2)^(n-1)

a_n  =  2^(n-1)

a1  =  2^(1-1)

      =   2^(0)

      =   1

a2  =  2^(2-1)

      =   2^(1)

      =   2

a3  =  2^(3-1)

      =   2^(2)

      =   4

a4  =  2^(4-1)

      =   2^(3)

      =   8

a5  =  2^(5-1)

      =   2^(4)

      =   16

The GP is 1, 2 , 4, 8, 16

Answer :

1)  

The n th term of a geometric sequence is a_n  =  0.8(-5)^(n-1)

The GP is 0.8, -4, 20, 100, 500

2)

The n th term of a geometric sequence is a_n  =  2^(n-1)

The GP is 1, 2 , 4, 8, 16.