# 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.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6