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