# Maths

a geometric progression has the second term as 9 and the fourth term as 81. find the sum of the first four terms.?

1. 👍 0
2. 👎 0
3. 👁 853
1. a1 = first term

r = common ratio

Sn = nth partial sum of a geometric sequence

an = a1 ∙ r ⁿ⁻¹

a2 = a1 ∙ r ²⁻¹ = a1 ∙ r¹ = a1 ∙ r = 9

a4 = a1 ∙ r ⁴⁻¹ = a1 ∙ r³ = 81

a4 / a2 = a1 ∙ r³ / a1 ∙ r = r²

r² = 81 / 9 = 9

r = √ r²

r = ± √ 9

r = ± 3

If:

a2 = a1 ∙ r = 9

then

a1 = 9 / r

a1 = 9 / ± 3

a1 = ± 3

Sn = a1 ∙ ( 1 - rⁿ ) / ( 1 - r )

S4 = a1 ∙ ( 1 - r⁴ ) / ( 1 - r )

There are 4 possible cases:

1.

a1 = 3 , r = 3

2.

a1 = 3 , r = - 3

3.

a1 = - 3 , r = 3

4.

a1 = - 3 , r = - 3

__________________________________________
Remark:
( - 3 )⁴ = ( - 3 )⁴ = [ ( - 1 ) ∙ 3 ]⁴ = ( - 1 )⁴ ∙ 3⁴ = 1 ∙ 3⁴ =3⁴

so
( - 3 )⁴ = 3⁴
___________________________________________

First case:

a1 = 3 , r = 3

S4 = a1 ∙ ( 1 - r⁴ ) / ( 1 - r )

S4 = 3 ∙ ( 1 - 3⁴ ) / ( 1 - 3 )

S4 = 3 ∙ ( 1 - 81 ) / ( - 2 )

S4 = 3 ∙ ( - 80 ) / ( - 2 )

S4 = 3 ∙ 40

S4 = 120

Second case:

a1 = 3 , r = - 3

S4 = a1 ∙ ( 1 - r⁴ ) / ( 1 - r )

S4 = 3 ∙ [ 1 - ( - 3)⁴ ] / [ 1 - ( - 3 ) ]

S4 = 3 ∙ ( 1 -- 3⁴ ) / [ 1 - ( - 3 ) ]

S4 = 3 ∙ ( 1 - 81 ) / ( 1 + 3 )

S4 = 3 ∙ ( - 80 ) / 4

S4 = 3 ∙ ( - 20 )

S4 = - 60

Third case:

a1 = - 3 , r = 3

S4 = a1 ∙ ( 1 - r⁴ ) / ( 1 - r )

S4 = - 3 ∙ ( 1 - 3⁴ ) / ( 1 - 3 )

S4 = - 3 ∙ ( 1 - 81 ) / ( - 2 )

S4 = - 3 ∙ ( - 80 ) / ( - 2 )

S4 = - 3 ∙ 40

S4 = - 120

Fourth case:

a1 = - 3 , r = - 3

S4 = a1 ∙ ( 1 - r⁴ ) / ( 1 - r )

S4 = - 3 ∙ [ 1 - ( - 3 )⁴ ] / [ 1 - ( - 3 ) ]

S4 = - 3 ∙ ( 1 - 3⁴ ) / ( 1 + 3 )

S4 = - 3 ∙ ( 1 - 81 ) / 4

S4 = - 3 ∙ ( - 80 ) / 4

S4 = - 3 ∙ ( - 20 )

S4 = 60

So:

S4 = ± 60

OR

S4 = ± 120

1. 👍 1
2. 👎 0

## Similar Questions

1. ### Arithmetic Sequence

Given the third term of an arithmetic sequence less than the fourth term by three. The seventh term is two times the fifth term. Find the common difference and the first term.

2. ### mathematics

A goemetric progression is such that the 3rd term is 9times the first term ,while the 2nd term is one-twenty fourth of the 5th term.Find the 4th term

3. ### Alg 2

The first term of a geometric sequence is 25, and the fourth term is 1\5. (a) Find the common ratio r. r = Find the fifth term. a5 = (b) Find the partial sum of the first eight terms. S8 =

4. ### Math

In a geometric sequence, the fourth term is -4, and the eighth term is -324. What is the tenth term?

1. ### uniuyo

The second and fifth term of a geometric progression are 21 and 567 respectively. Find the first term and the common ratio of the progression

2. ### Math

a geometric progression has a third term of 20 and a sum to infinity which is three times the first term. find the first term

3. ### quine

The 2nd and 5th term of a geometric progression are - 6 & 48 find the sum of the first four term respectively

4. ### math

Three numbers form a geometric progression. If the second term is increased by 2, then the progression will become arithmetic and if, after this, the last term is increased by 9, then the progression will again become geometric.

1. ### math

The 3rd term of an Arithmetic Progression is 10 more than the first term while the fifth term is 15 more than the second term. Find the sum of the 8th and 15th terms of the Arithmetic Progression if the 7th term is seven times the

2. ### Maths

A Geometric progression X has a first term of 4a and its sixth term is 256. Another Geometric progression Y has a first term of 3a and its fifth term is 48. Find the: (i) First term of X (ii) Sum of the first four terms of X

3. ### Maths

The 5th term of an arithmetic progression is 3times of 2nd term and 12th term exceeds 2times of 6th term by 1. Find the 16th term

4. ### math

The 3rd term of a geometric progression is nine times the first term.if the second term is one-twenty fourth the 5th term.find the 4th term. solution ar^2=9a r=sqr of 9 r=3.pls help me on how to get the first term(a)