FIND THE SUM OF GEOMETRIC SERIES

To find the sum of geometric series, we use the formula given below.

Based on the value of common ratio, we use one of the formula given below.

sn = arn - 1(r - 1), if r > 1sn = a1 - rn(1 - r), if r < 1sn = na, if r = 1

Use the formula for Sn 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

sn = 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

sn = 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

sn = 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

sn = 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

sn = 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

sn = a1(1 - rn)/(1 – r)

s15 = 26(1 - (-0.625)15)/1 + 0.625)

s15 = 26(1 - (-0.625)15)/(1.625)

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