The first term of GP is - 3 and square of second term is equal to the 4th term.Find 7th term

Answer 1

In a geometric progression:

a1 = initial value

r = common ratio

the n-th term:

an = a1 ∙ r^(n-1)

In this case:

a1 = - 3

a2= a1 ∙ r^(2-1) = ( - 3 ) ∙ r^1 = - 3 r

a4 = a1 ∙ r^(4-1) = ( - 3 ) ∙ r^3 = - 3 r^3

Square of second term is equal to the 4th term mean:

a2^2 = a4

( - 3 r )^2 = - 3 r^3

9 r ^2 = - 3 r^3

Divide both sides by 9 r ^2

1 = - 3 r^3 / 9 r ^2

1 = - 3 r^2 ∙ r / 3 r ^2 ∙ 3

1 = - r / 3

Multiply both sides by - 3

- 3 = r

r = - 3

a7 = a1 ∙ r^ ( 7 - 1 ) = ( - 3 ) ∙ r^6 = ( - 3 ) ∙ ( - 3 ) ^6 = ( - 3 ) ∙ 729 = -2187

Proof:

a2 = - 3 r = ( - 3 ) ∙ ( - 3 ) = 9

a4 = - 3 r^3 = ( - 3 ) ∙ ( - 3 )^3 = ( - 3 ) ∙ ( - 27 ) = 81

a4 = a2^2