NARRATOR: Exercise 2.
Find the limiting sum for the geometric series
3 on 2 plus 9 on 8 plus 27 on 32 plus, etc.
In this geometric series, the first term is 3 on 2.
So a is equal to 3 on 2.
The common ratio, r, can be determined
by dividing consecutive terms in the series.
9 on 8 divided by 3 on 2
is equivalent to 9 and 8 multiplied by two-thirds,
which is 18 on 24 or three-quarters.
Similarly, 27 divided by 32 divided by 9 on 8
is equal to 27 on 32 times eight-ninths,
which is also three-quarters.
And so we establish a common ratio, r equals three-quarters.
The limiting sum of a geometric series
is S infinity which is equal to a divided by 1 minus r.
In this case, a is 3 on 2 and r is three-quarters.
So the limiting sum is 3 on 2 divided by 1 minus three-quarters,
that is 3 on 2 divided by a quarter
or 3 on 2 multiplied by 4,
which is 12 on 2 or 6.
So the limiting sum
for the geometric series is 6.