Answer :
The first term of the given sequence (a) = 6561
Step-by-step explanation:
Let the first term = a and common difference = d
Given,
[tex]a_{3}[/tex] = 729 and [tex]a_{4}[/tex] = 243
To find, the first term of the given sequence (a) = ?
We know that,
The nth term of a G.P.
[tex]a_{n} =ar^{n-1}[/tex]
The 3rd term of a G.P.
[tex]a_{3} =ar^{3-1}[/tex]
⇒ [tex]ar^{2}[/tex] = 729 ..............(1)
The 4th term of a G.P.
[tex]a_{4} =ar^{4-1}[/tex]
⇒ [tex]ar^{3}[/tex] = 243 ..............(2)
Dividing equation (2) by (1), we get
[tex]\dfrac{ar^{3}}{ar^{2}}[/tex] = [tex]\dfrac{243}{729}[/tex]
⇒ [tex]r=\dfrac{1}{3}[/tex]
Put [tex]r=\dfrac{1}{3}[/tex] in equation (1), we get
[tex]a(\dfrac{1}{3})^{2}[/tex] = 729
⇒ [tex]a(\dfrac{1}{9})[/tex] = 729
⇒ a = 9 × 729 = 6561
∴ The first term of the given sequence (a) = 6561