The above formulas do not hold for r = 1. For r = 1, the sum of n terms of the Geometric Progression is Sn
n
= na.
(ii)When the numerical value of r is less than 1 (i.e., - 1 < r < 1), then the formula Sn
n
= a(1−rn)(1−r)
(
1
−
r
n
)
(
1
−
r
)
is used.
(iii) When the numerical value of r is greater than 1 (i.e., r > 1 or, r < -1) then the formula Sn
n
= a(rn−1)(r−1)
(
r
n
−
1
)
(
r
−
1
)
is used.
(iv) When r = 1, then Sn
n
= a + a + a + a + a + .................... to n terms = na.
(v) If l is the last term of the Geometric Progression, then l = arn−1
n
−
1
.
Therefore, Sn
n
= a(1−rn1−r
1
−
r
n
1
−
r
) = (a−arn1−r
a
−
a
r
n
1
−
r
) = a−(arn−1)r(1−r)
a
−
(
a
r
n
−
1
)
r
(
1
−
r
)
= a−lr1−r
a
−
l
r
1
−
r
Thus, Sn
n
= a−lr1−r
a
−
l
r
1
−
r
Or, Sn
n
= lr−ar−1
l
r
−
a
r
−
1
, r ≠ 1.
