The first term of a finite GP series is a positive number. If the sum of the terms of the GP series is negative . Then find out the number of term of the GP series
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The first term of a finite GP series is a positive number. If the sum of the terms of the GP series is negative . Then find out the number of term of the GP series
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Step-by-step explanation:
Let a,ar,ar
2
be the terms of G.P
a=ar+ar
2
....[Given]
⇒r
2
+r−1=0
We have a formula for solving quadratic equation ax
2
+bx+c=0 is
x=
2a
−b±
b
2
−4ac
⇒r=
2
−1±
5
a negative value will be discarded because the terms in the question are positive.
∴r=
2
−1+
5
Answer:
Hey mate, here is your answer.
Step-by-step explanation:
Sum of the First n Terms of a Geometric Sequence
If a sequence is geometric there are ways to find the sum of the first n terms, denoted Sn, without actually adding all of the terms.
To find the sum of the first Sn terms of a geometric sequence use the formula
Sn=a1(1−rn)1−r,r≠1,
where n is the number of terms, a1 is the first term and r is the common ratio.
The sum of the first n terms of a geometric sequence is called geometric series.
Example 1:
Find the sum of the first 8 terms of the geometric series if a1=1 and r=2.
S8=1(1−28)1−2=255
Example 2:
Find S10 of the geometric sequence 24,12,6,⋯.
First, find r.
r=r2r1=1224=12
Now, find the sum:
S10=24(1−(12)10)1−12=306964
Example 3:
Evaluate.
∑n=1103(−2)n−1
(You are finding S10 for the series 3−6+12−24+⋯, whose common ratio is −2.)
Sn=a1(1−rn)1−rS10=3[1−(−2)10]1−(−2)=3(1−1024)3=−1023
In order for an infinite geometric series to have a sum, the common ratio r must be between −1 and 1. Then as n increases, rn gets closer and closer to 0. To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, S=a11−r, where a1 is the first term and r is the common ratio.
Example 4:
Find the sum of the infinite geometric sequence
27,18,12,8,⋯.
First find r:
r=a2a1=1827=23
Then find the sum:
S=a11−r
S=271−23=81
Example 5:
Find the sum of the infinite geometric sequence
8,12,18,27,⋯ if it exists.
First find r:
r=a2a1=128=32
Since r=32 is not less than one the series has no sum.
There is a formula to calculate the nth term of an geometric series, that is, the sum of the first n terms of an geometric sequence.
See also: sigma notation of a series and sum of the first n terms of an arithmetic sequence
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