A geometric sequence , the fifth term is four times the third terms, the second term is 4.

Question

If r<0, determine:
A) the value of r, the common ratio.
B) the value of a.
C) the tenth term.
D) the sum of the 15 terms.

Answer 1

You have a GP where 2nd term is 4 -----> ar=4
the fifth term is four times the third term ---- ar^4 = ar^2

ar^4 = ar^2
divide by ar^2, r can't be zero
r^2 = 1
r = ± 1

then in ar=4
if r = +1, a = 4
if r = -1, a = -4

Do C and D by simply using the formulas that you learned.

Answer 2

In geometric sequence:

an = a1 ∙ r ⁿ ⁻ ¹

a2 = a1 ∙ r

a3 = a1 ∙ r ²

a5 = a1 ∙ r ⁴

The fifth term is four times the third terms mean; a5 = 4 ∙ a3

a1 ∙ r ⁴ = 4 ∙ a1 ∙ r ²

Divide both sides by ( a1 ∙ r ² )

r ² = 4

r = ± √4

r = ± 2

In this case r < 0 so:

r = - 2

The second term is 4 mean; a2 = 4

a1 ∙ r = 4

a1 ∙ ( - 2 ) = 4

Divide both sides by - 2

a1 = 4 / - 2

a1 = - 2

a10 = a1 ∙ r⁹

a10 = ( - 2 ) ∙ ( - 2 )⁹

a10 = ( - 2 ) ∙ ( - 512 )

a10 = 1024

Sn = a1 ( 1 − r ⁿ ) / ( 1 − r )

In this case n = 10

S10 = ( - 2 ) ∙ [ 1 − ( - 2 )¹⁰ ) / [ 1 − ( - 2 ) ]

S10 = ( - 2 ) ∙ ( 1 − 1024 ) / ( 1 + 2 )

S10 = ( - 2 ) ∙ ( − 1023 ) / 3

S10 = 2046 / 3

S10 = 682

Answer 3

The sum of the 15 terms.

Sn = a1 ( 1 − r ⁿ ) / ( 1 − r )

where n = 15

S15 = ( - 2 ) ∙ [ 1 − ( - 2 )¹⁵ ] / [ 1 − ( - 2 ) ]

S15 = ( - 2 ) ∙ [ 1 − ( - 32 768 ) ] / ( 1 + 2 )

S15 = ( - 2 ) ∙ ( 1 + 32 768 ) / 3

S15 = ( - 2 ) ∙ 32 769 / 3

S15 = - 65 538 / 3

S15 = - 21 846‬