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?

Answers

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

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