Find all geometric sequences such that the sum of the first two terms is 24 and the sum of the first three terms is 26.

The nth term Tn = - 8 (-1/4) ^ (n-1) or Tn = 6 (1/3) ^ (n-1) can be used to find all geometric sequences

Step-by-step explanation:

Let the first three terms be a/r, a, ar ... where a is the first term and r is the common ratio of the geometric sequence.

If the sum of the first two term is 24, then a/r + a = 24 ... (1)

and the sum of the first three terms is 26 ... then a/r+a+ar = 26 ... (2)

Substtituting equation 1 into 2 we have;

24+ar = 26

ar = 2

a = 2/r ... (3)

Substituting a = 2/r into equation 1 will give;

(2/r)) / r+2/r = 24

2/r²+2/r = 24

(2+2r) / r² = 24

2+2r = 24r²

1+r = 12r²

12r²-r-1 = 0

12r²-4r+3r - 1 = 0

4r (3r-1) + 1 (3r-1) = 0

(4r+1) (3r-1) = 0

r = - 1/4 0r 1/3

Since a = 2/r then a = 2 / (-1/4) or a = 2 / (1/3)

a = - 8 or 6

All the geometric sequence can be found by simply knowing the formula for heir nth term. nth term of a geometric sequence is expressed as

if r = - 1/4 and a = - 8

Tn = - 8 (-1/4) ^ (n-1)

if r = 1/3 and a = 6

Tn = 6 (1/3) ^ (n-1)

The nth term of the sequence above can be used to find all the geometric sequence where n is the number of terms