Question

Given a G.P. with a = 729 and 7th term 64, determine S7.

Answer 1

a = 729
a7 = 64
Let r be the common ratio of the G.P.
It is known that, an = a r n-1
a7 = ar7-1 = (729)r 6
⇒ 64 = 729 r 6
$⇒ r^6 = \frac{64 }{ 729}$
$⇒ r^6 = (\frac{2 }{ 3})^6$
$⇒ r =\frac{ 2 }{ 3}$
Also, it is known that, $S_n = \frac{a(1 - r^n) }{ 1 - r}$
∴$S_7 = \frac{729 [ 1 - (\frac{2 }{ 3})^7 ] }{ 1 - \frac{2 }{3}}$
$= 3 × 729 [1 - (\frac{2 }{ 3})^7]$
$= (3)^ [ \frac{(3)^7 - (2)^7 }{ (3)^7} ]$
$= (3)^7 - (2)^7$
$= 2187 - 128$
$= 2059$