Find the Sum of n Terms From the Given Terms of the Geometric Series :
In this section, we will learn how to find the sum of n terms from the given terms of a geometric series.
Example 1 :
The second term of the geometric series is 3 and the common ratio is 4/5. Find the sum of first 23 consecutive terms in the given geometric series.
Second term (t2) = 3
ar = 3 r = 4/5 < 1 and n = 23
a(4/5) = 3
a = (3 ⋅ 5)/4
a = 15/4
Sum of first 23 terms = a (1 - rn)/(1 - r)
= 15/4 (1 - (4/5)23)/(1 - (4/5))
= 15/4 [ (1 - (4/5)23)/(1/5) ]
= (15/4) ⋅ (5/1) (1-(4/5)^23)
= (75/4) (1-(4/5)^23)
Example 2 :
Suppose that five people are ill during the first week of an epidemic and each sick person spreads the contagious disease to four other people by the end of the second week and so on. By the end of 15th week, how many people will be affected by the epidemic?
By writing the given data as sequence, we get
5, 5(4), 5(4)², ..............
5, 20, 80, ..............
Since we find total number of people affected by the epidemic the above sequence is going to be changed as
5 + 20 + 80 + ..............
here a = 5 r = 20/5 ==> 4 and n = 15
Sn = a (rn-1)/(r-1)
S15 = 5 (415 - 1)/(4 - 1)
S15 = (5/3) (415 - 1)
Example 3 :
A gardener wanted to reward a boy for his good deeds by giving some mangoes. He gave the boy two choices. He could either have 1000 mangoes at once or he could get 1 mango on the first day, 2 on the second day, 4 on the third day, 8 mangoes on the fourth day and so on for ten days. Which option should the boy choose to get the maximum number of mangoes ?
By choosing the first option, he will get 1000 mangoes.
To find the total number of mangoes that he is going to get by choosing the second option, we should find the sum of first 10 terms of the series.
Writing the second option as series, we get
1 + 2 + 4 + 8 + ....... 10 days
Here a = 1 r = 2 > 1
sn = a (rn - 1) / (r - 1)
= 1 (210 - 1)/(2 - 1)
= 1024 - 1
By choosing the second way, he will get 1023 mangoes.
Therefore the boy should choose the second way to get the maximum number of mangoes.
Example 4 :
A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 9 and the sum of the last two terms is 36. Find the series.
Let a, ar, ar2 and ar³ be the first four terms of geometric series
Sum of the first two terms = 9
a + ar = 9
a(1 + r) = 9 ----(1)
Sum of the last two terms = 36
ar2 + ar3 = 36
ar2 (1 + r) = 36 --- (2)
Substitute a (1 + r) = 9 in (2), we get
r2 (9) = 36
r2 = 36/9
r2 = 4
r = √4
r = ± 2
r = -2 is not admissible. So, the value of r is 2.
By applying the value of r in (1), we get
a (1 + 2) = 9
a(3) = 9
a = 3
By applying the values of a and r, we get
3 + 3(2) + 3(2)2 + 3(2)3+ .........
Therefore, the series is
3 + 6 + 12 + 24 + ......
After having gone through the stuff given above, we hope that the students would have understood how to find the sum of n terms from the given terms of the geometric series.
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