GEOMETRIC SERIES WORKSHEET

Question 1 :

Find the sum of the following geometric series.

1 + 3 + 9 + .......... to 10 terms

Question 2 :

Find the sum of 8 terms of the following geometric series.

1 + (-3) + 9 + (-27) +..........

Question 3 :

Find the sum of 30 terms of the following geometric series.

7 + 7 + 7 + 7 +..........

Question 4 :

Find the sum of terms of the following geometric series.

1 + 2 + 4 + 8 +.......... + 2048

Question 5 :

Find the sum of the following infinite geometric series.

3 + 1 + 1/3 +.......... ∞

Question 6 :

Find the sum of the following infinite geometric series.

-3 + (-12) + (-48) +.......... ∞

Question 7 :

A ball is dropped from a height of 6 m and on each bounce it bounces 2/3 of its previous height.

(i) What is the total length of the downward paths?

(ii) What is the total length of the upward paths?

(iii) How far does the ball travel till it stops bouncing?

1. Answer :

1 + 3 + 9 + .......... to 10 terms

a₁ = 1

r = a₂/a₁ = 3/1 = 3 > 1

n = 10

Formula to find sum of n terms of a geometric series where r ≠ 1.

S_n = a₁(rⁿ - 1)/(r - 1)

Substitute a₁ = 1 and r = 3.

S₁₀ = 1(3¹⁰ - 1)/(3 - 1)

= (59049 - 1)/2

= 59048/2

= 29524

2. Answer :

1 + (-3) + 9 + (-27) +.......... to 8 terms

a₁ = 1

r = a₂/a₁ = -3/1 = -3 < 1

n = 10

Formula to find sum of n terms of a geometric series where r > 1.

S_n = a₁(rⁿ - 1)/(r - 1)

Substitute a₁ = 1, r = -3 and n = 8.

S₈ = 1[(-3)⁸ - 1]/(-3 - 1)

= (6561 - 1)/(-4)

= 6560/(-4)

= -1640

3. Answer :

7 + 7 + 7 + 7 +.......... to 30 terms

a₁ = 7

r = a₂/a₁ = 7/7 = 1

n = 30

Formula to find sum of n terms of a geometric series where r = 1.

S_n = na₁

Substitute a₁ = 7 and r = 1.

S₃₀ = 30(7)

= 210

4. Answer :

1 + 2 + 4 + 8 +.......... + 2048

a₁ = 1

r = a₂/a₁ = 2/1 = 2 > 1

The value of n is not given. So, we have to find the value of n.

Let 2048 be the n^th term.

a_n = 2048

a₁r^n-1 = 2048

Substitute a₁ = 1 and r = 2.

1(2)^n-1 = 2048

2^n-1 = 2¹¹

n - 1 = 11

n = 12

Formula to find sum of n terms of a geometric series where r ≠ 1.

S_n = a₁(rⁿ - 1)/(r - 1)

Substitute a₁ = 1, r = -3 and n = 12.

S₁₂ = 1[2¹² - 1]/(2 - 1)

= (4096 - 1)/1

= 4095

5. Answer :

3 + 1 + 1/3 +.......... ∞

a₁ = 3

r = a₂/a₁ = 1/3

The value of r (= 1/3) is in the interval -1 < r < 1.

The sum for the given infinite geometric series exists.

S_∞ = a₁/(1 - r)

Substitute a₁ = 3 and r = 1/3.

S_∞ = 3/(1 - 1/3)

= 3/(2/3)

= 3(3/2)

= 9/2

6. Answer :

-3 + (-12) + (-48) +.......... ∞

a₁ = 3

r = a₂/a₁ = -12/(-3) = 4

The value of r (= 4) is not in the interval -1 < r < 1.

So, the sum for the given infinite geometric series does not exist.

7. Answer :

(i) Distance covered in the downward path :

= 6 + 4 + 8/3 + 16/9 + .............

a₁ = 6

r = a₂/a₁ = 4/6 = 2/3

The value of r (= 2/3) is in the interval -1 < r < 1.

Formula to find sum of infinite geometric series :

S_∞ = a₁/(1 - r)

Substitute a₁ = 6 and r = 2/3.

= 6/(1 - 2/3)

= 6/(1/3)

= 6(3/1)

= 18 m

(ii)  Distance covered in the upward path :

= 4 + 8/3 + 16/9 + ............

S_∞ = a₁/(1 - r)

Substitute a₁ = 4 and r = 2/3.

S_∞ = 4/(1 - 2/3)

= 4/(1/3)

= 4(3/1)

= 12 m

(iii) Total distance covered :

= 18 + 12

= 30 m

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic 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 fractions

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