GEOMETRIC SEQUENCE AND SERIES WORKSHEET

1.  A geometric sequence has first term 3 and common ratio -2. Find the 10^th term.

2.  The second term of a geometric sequence is 6 and the fourth term is 96. Find the possible values of the first term and the common ratio.

3.  Find the number of terms in the geometric sequence :

1, 2, 4, 8, ......., 512

4.  Find the sum of 10 terms of the geometric sequence : 

1, 2, 4, 8, ..........

5.  Find the sum of first 8 terms of a geometric sequence whose n^th term 3^2n-1.

6.  Find the first term of a geometric sequence whose common ratio is 5 and sum to first 6 terms is 46872.

7.  Find the sum of the geometric series :

2 + 6 + 18 + ........ + 13122

8.  Find the sum of the geometric series :

5 + 5 + 5 + ........ to 27 terms

9.  Find the sum to infinity of 9 + 3 + 1 + ........

10.  Peterson writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with the instruction that they continue the process similarly. Assuming that the process is unaltered and it costs $2 to mail one letter, find the amount spent on postage when 8^th set of letters is mailed.

1. Answer :

Formula for n^th term of a geometric sequence : 

 a_n = a₁r^{n - 1}

Substitute n = 10, a₁ = 3 and d = -2.

a₁₀ = 3(-2)^{10 - 1}

= 3(-2)⁹

= 3(-512)

= -1536

2. Answer :

a₂ = 6 and a₄ = 96

(2) ÷ (1) :

r² = 16

Take square root on both sides. 

r = ±4

The possible values of the common ratio are -4 and 4.

The possible values of the first term are -3/2 and 3/2.

3. Answer :

1, 2, 4, 8, ......., 512

This is a geometric sequence with the first term 1 and common ratio 2. 

Let a_n = 512.

 a₁r^{n - 1} = 512

Substitute a₁ = 1 and r = 2. 

1(2)^{n - 1} = 512

Write 512 as a power of 2.

2^{n - 1} = 2⁹

n - 1 = 9

n = 10

4. Answer :

1, 2, 4, 8, ..........

This is a geometric sequence with a₁ = 1 and r = 2.

Formula for the sum of first n terms of a geometric sequence.

S_n = a₁(1 - rⁿ)/(1 - r)

Substitute n = 10, a₁ = 1 and r = 2.

S₁₀ = 1(1 - 2¹⁰)/(1 - 2)

= 1(1 - 1024)/(-1)

= -1023/(-1)

= 1023

5. Answer :

a_n = 3^2n-1

Common ratio :

r = a₂/a₁

r = 27/3

r = 9

Formula for the sum of first n terms of a geometric sequence.

S_n = a₁(1 - rⁿ)/(1 - r)

Substitute n = 8, a₁ = 3 and r = 9.

S₈ = 3(1 - 3⁸)/(1 - 3)

= 3(1 - 6561)/(-2)

= 3(-6560)/(-2)

= 3(3280)

= 9840

6. Answer :

S₆ = 46872

a₁(1 - r⁶)/(1 - r) = 46872

Substitute r = 5.

a₁(1 - 5⁶)/(1 - 5) = 46872

a₁(1 - 15625)/(-4) = 46872

a₁(-15624)/(-4) = 46872

3906a₁ = 46872

Divide each side by 3906.

a₁ = 12

7. Answer :

2 + 6 + 18 + ........ + 13122 

This is a geometric series with a₁ = 2 and r = 3.

Let a_n = 13122.

a_n = 13122

a₁r^{n - 1} = 13122

Substitute a₁ = 2 and r = 3.

2(3)^{n - 1} = 13122

3^{n - 1} = 6561

Write 6561 as a power of 3. 

3^{n - 1} = 3⁸

n - 1 = 8

n = 9

Formula for the sum of first n terms of a geometric sequence.

S_n = a₁(1 - rⁿ)/(1 - r)

Substitute n = 9, a₁ = 2 and r = 3.

S₉ = 2(1 - 3⁹)/(1 - 3)

= 2(1 - 19683)/(-2)

= 2(-19682)/(-2)

= 19682

8. Answer :

5 + 5 + 5 + ........ to 27 terms 

This is a geometric series with a₁ = 5 and r = 1.

S_n = na₁

Substitute n = 27 and a₁ = 5.

S_n = 27(5)

= 135

9. Answer :

9 + 3 + 1 + ........

This is a geometric series with a₁ = 9 and r = 1/3.

S_∞ = a₁/(1 - r)

Substitute a₁ = 9 and r = 1/3.

S_∞ = 9/(1 - 1/3)

= 9/(2/3)

= 9 ⋅ 3/2

= 27/2

10. Answer :

Amount spent when the first set of letters is mailed :

=  4 ⋅ 2

= $8

Amount spent when the second set of letters is mailed :

=  4 ⋅ 4 ⋅ 2

= $32

Amount spent when the third set of letters is mailed :

=  4 ⋅ 4 ⋅ 4 ⋅ 2

= $128

If this pattern continues, we will have a geometric sequence with the first term 8 and common ratio 4 as shown below. 

8, 32, 128, ............. to 8 terms

Find the sum of the terms in the above geometric sequence. 

S_n = a₁(1 - rⁿ)/(1 - r)

Substitute n = 8, a₁ = 8 and r = 4.

S₈ = 8(1 - 4⁸)/(1 - 4)

= 8(1 - 65536)/(-3)

= 8(-65535)/(-3)

= 8(21845)

= $174,760

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