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.
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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