Question

A geometric series has a common ratio of (-2) and the first term is 3.

Answer 1

EXPLANATION.

Geometric series.

Common ratio = - 2.

First term = 3.

As we know that,

If a is the first term and r be the common ratio of G.P.

⇒ a, ar, ar², ar³, ar⁴, . . . . .

⇒ Tₙ = a rⁿ⁻¹.

First term = a = 3.

Common ratio = r = - 2.

⇒ (3), [(3)(-2)], [(3)(-2)²], [(3)(-2)³], . . . . .

Series = (3), (-6), (12), (-24), . . . . .

                                                                                                                   

MORE INFORMATION.

Sum of Geometric progression.

⇒ Sₙ = [a(1 - rⁿ)/(1 - r)]  r < 1.

⇒ Sₙ = [a(rⁿ - 1)/(r - 1)]  r > 1.

Sum of infinite terms of geometric progression.

⇒ |r| < 1.

⇒ - 1 < r < 1.

⇒ Sₙ = [a(1 - rⁿ)/(1 - r)]  r < 1.

n tends to ∞.

⇒ S(∞) = [a/(1 - r)].

If a, b and c are in G.P.

⇒ b² = ac.

Answer 2

Answer:

Question :-

• A geometric series has a common ratio of -2 and the first term is 3.

$\mathcal\purple{Given -}$

• A geometric series and given
• Common difference =-2
• First term=3

To find :-

• The geometric series.

$\mathcal\purple{Explanation : -}$

• Here we comes to know that,

• a is first term and r be the common difference of geometric progression.

• a,ar^2,ar^3,ar^4..

• $Tn = a {r}^{n - 1}$
• Here we should findthe geometric series so lets take , a=3 and r =-2.

• $a = 3$

• $ar = 3 \times - 2 = - 6$
• $a {r}^{2} = 3 \times { - 2}^{2} = 12$
• $ar {}^{3} = 3 \times { - 2}^{3} = - 24$

♧Therefore ,

• 3,-6,12,-24 are the geometric series.

♧Hope it helps u mate.

♧Thank you .