Question

Find the 12th term of a G.P whose 8th term is 192 and the common ratio is 2

Answer 1

If a₁ is the first term, $a_{n}$ is the nth term and r is the common ratio of the given G.P. series then
$a_{n}$=a₁rⁿ⁻¹
a₈=a₁×(2)⁸⁻¹=192
or, a₁×2⁷=192
or, a₁=192/2⁷
or, a₁=2⁶×3/2⁷
or, a₁=3/2
∴, a₁₂=a₁r¹²⁻¹
=3/2×(2)¹¹
=3×(2)¹⁰
=3×1024
=3072

Answer 2

Solution:-
Given; Common ratio = 2, 8th term = 192
Let a be the 1st term of the G.P.
We know that,
a(n) = ar⁽ⁿ⁻¹⁾
where a(n) = nth term of G.P.
n = number of terms
a = first term
r = common ratio
a₈ = ar⁽⁸⁻¹⁾
a₈ = ar⁷
ar⁷ = 192
a(2)⁷ = 192
a(2)⁷ = (2)⁶ (3)
⇒ a = {(2)⁶ (3)}/(2)⁷
⇒ a = {(3)/(2)⁷⁻⁶}
⇒ a = (3)/(2)¹
⇒ a = 3/2  (first term)
So,
n₁₂ = ar⁽¹²⁻¹⁾
n₁₂ = 3/2 × 2⁽¹²⁻¹⁾
n₁₂ = 3/2 × 2⁽¹¹⁾
n₁₂ = {3 × (2)⁽¹¹⁻¹⁾}
n₁₂ = 3 × 2⁽¹⁰⁾
n₁₂ = 3 × 1024
n₁₂ = 3072
Answer.