GEOMETRIC SEQUENCE AND SERIES WORKSHEET

1.  A geometric sequence has first term 3 and common ratio -2. Find the 10th 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 nth term 32n-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 8th set of letters is mailed. Formula for nth term of a geometric sequence :

an = a1rn - 1

Substitute n = 10, a1 = 3 and d = -2.

a10 = 3(-2)10 - 1

3(-2)9

= 3(-512)

= -1536

a2 = 6 and a4 = 96

 a2 = 6a1r2 - 1 = 6a1r = 6 ----(1) a4 = 96a1r4 - 1 = 96a1r3 = 96 ----(2)

(2) ÷ (1) :

r2 = 16

Take square root on both sides.

r = ±4

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

 Substitute r = -4 in (1). a1(-4) = 6a1 = -3/2 Substitute r = 4 in (1). a1(4) = 6a1 = 3/2

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

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

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

Let an = 512.

a1rn - 1 = 512

Substitute a1 = 1 and r = 2.

1(2)n - 1 = 512

Write 512 as a power of 2.

2n - 1 = 29

n - 1 = 9

n = 10

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.

Sn = 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

an = 32n-1

a1 = 32(1) - 1

= 32 - 1

= 31

= 3

a2 = 32(2)-1

= 34 - 1

= 33

=  27

Common ratio :

r = a2/a1

r = 27/3

r = 9

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

Sn = 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

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

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

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

Let a= 13122.

an = 13122

a1rn - 1 = 13122

Substitute a= 2 and r = 3.

2(3)n - 1 = 13122

3n - 1 = 6561

Write 6561 as a power of 3.

3n - 1 = 38

n - 1 = 8

n = 9

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

Sn = 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

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

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

Sn = na1

Substitute n = 27 and a= 5.

Sn = 27(5)

= 135

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

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

S∞ = a1/(1 - r)

Substitute a= 9 and r = 1/3.

S∞ = 9/(1 - 1/3)

= 9/(2/3)

= 9  3/2

= 27/2

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 ⋅ 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.

Sn = 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 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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