Question

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

Answer 1

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 2

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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