4th term = ar4-1 = ar3 = \(\frac{1}{27}\)
7th term = ar7-1= ar6 = \(\frac{1}{729}\)
Dividing the 7th term by the 4th term
\(\frac{ar^6}{ar^3} = \cfrac{\frac{1}{729}}{\frac{1}{27}}\)
⇒ r3 = \(\frac{1}{27}\) ........... (1)
∴ r = \(\frac{1}{3}\)
ar3 = \(\frac{1}{27}\) [Putting from eqn (i)]
a\(\frac{1}{27}\) = \(\frac{1}{27}\)
∴ a = 1
Sum of a G.P. series is represented by the formula, Sn = \(a\frac{1 - r^n}{1-r}\), when |r|<1.
‘Sn’ represents the sum of the G.P. series up to nth terms,
‘a’ represents the first term,
‘r’ represents the common ratio and
‘n’ represents the number of terms.
Here,
a = 1
r = \(\frac{1}{3}\)
n terms