FIND THE SUM OF GEOMETRIC SERIES

Use the formula for S_n to calculate the sum of following geometric series.

Problem 1 :

5 + 10 + 20 + … to 6 terms

Solution :

5 + 10 + 20 + … to 6 terms

a₁ = 5, n = 6

r = 10/5

r = 2 if r > 1

s_n = a₁(rⁿ – 1)/(r – 1)

= 5(2⁶ – 1)/(2 – 1)

= 5(64 – 1)/1

= 5(63)

S₆ = 315

Problem 2 :

4 + 12 + 36 + … to 10 terms

Solution :

4 + 12 + 36 + … to 10 terms

a₁ = 4

n = 10

r = 12/4

r = 3 if r > 1

s_n = a₁(rⁿ – 1)/(r – 1)

= 4(3¹⁰ – 1)/(3 – 1)

= 4(3¹⁰ – 1)/2

S₁₀ = 2(3¹⁰ – 1)

Problem 3 :

1/3 + 1/6 + 1/12 + … to 8 terms

Solution :

1/3 + 1/6 + 1/12 + … to 8 terms

a₁ = 1/3, n = 8

r = (1/6) / (1/3), r = 1/2 if r < 1

s_n = a₁(1 – rⁿ) / (1 – r)

= 1/3(1 – (1/2)⁸) / (1 – 1/2)

= 1/3((1 – (1/2)⁸) / (2 – 1)/2

= 1/3(1 - 1/256) / (1/2)

= 1/3((256 – 1)/256) / (1/2)

= 1/3(255/256) / ((1/2)

S₈ = 85/128

Problem 4 :

100 - 20 + 4 + … to 20 terms

Solution :

100 - 20 + 4 + … to 20 terms

a₁ = 100, n = 20

r = -20/100

r = -1/5 if r < 1

s_n = a₁(1 – rⁿ)/(1 – r)

= 100(1 – (-1/5)²⁰)/(1 + 1/5)

s₂₀ = 100(1 – (-1/5)²⁰)/(6/5)

Problem 5 :

16 + 17.6 + 19.36 + … to 50 terms

Solution :

16 + 17.6 + 19.36 + … to 50 terms

a₁ = 16, n = 50

r = 17.6/16

r = 1.1 if r >1

s_n = a₁(rⁿ – 1)/(r – 1)

= 16((1.1)⁵⁰ – 1)/(1.1 – 1)

= 16((1.1)⁵⁰ – 1)/(0.1)

= 16((1.1)⁵⁰ – 1) × 10/(0.1) × 10

= 16((11)⁵⁰ – 10)

Problem 6 :

26 – 16.25 + 10. 15625  … to 15 terms

Solution :

a₁ = 26, n = 15

r = -16.25/26

r = -0.625 if r <1

s_n = a₁(1 - rⁿ)/(1 – r)

s₁₅ = 26(1 - (-0.625)¹⁵)/1 + 0.625)

s₁₅ = 26(1 - (-0.625)¹⁵)/(1.625)