Helpful for students

1. Geometric Progression(G.P.)

2. A sequence is said to be in Geometric
Progression if the ratio of each term to its
preceding one is the same throughout the
whole sequence. The constant ratio is called the
common ratio (c. r.) or simply ( r).
For example:
(i) 3, 9, 27, ..……..
(ii) 22, 23, 24, ……..
(iii) -1, 2, -4, 8......... etc.

3. nth term of a G. P.
Let a be the first term and r be the common
ratio. Then
The first term, t1= a = ar0 = ar1–1
The second term, t2 = ar = ar2–1
The third term, t3 = ar2 = ar 3–1
…………………………….
The nth term, tn = arn–1
 nth term= tn = arn–1

4. Sum of the n terms of the series
in G.P.
Let a be the first term, r the common ratio, and
tn the last term of G.P. Then the series of G.P. is
in the form
a + ar + ar2 + …… + arn–1.
Let, Sn denote the sum of the series. Then
Sn = a + ar + ar2 +…………..+ arn–1 – – – – – – – (i)
Multiplying this series by r on both sides
rSn = ar + ar2 + ar3 + ……………… + arn – – – – (ii)

5. Subtracting the series (ii) from (i)
(1 –r) Sn = a – arn
Sn =
a(1–rn
)
1–r
Sn =
a(1–rn
)
1–r
ifr<1
=
a(rn
–1)
r–1
ifr>1
Iflbethelasttermoftheseries,thenitcanalsobewrittenas.
Sn =
a(rn
–1)
r–1
=
arn
–a
r–1
=
arn–1
.r–a
r–1
=
lr–a
r–1

6. Sum of an infinite Geometric
Series.
An infinite geometric series will have a sum if
the numerical value of the common ratio r is less
than 1, i.e. | r | < 1
Then formula becomes
S =
𝑎
1−𝑟
When, r < 1.
But when r  1, the sum becomes infinite, which
has no meaning

7. Geometric mean
If three numbers are in G. P., the middle term is
called the 'Geometric mean' between the other
two.
Let a, G, b be the three numbers in G. P; then
common ratio of two consecutive numbers is the
same
i.e. Ga = bG or, G2 = ab ...𝐺 = √ ab
... G.M.= √ ab
Hence, the geometric mean of two numbers a
and b is the square root of their product i.e. √ ab

8. Geometric means between two
numbers.
Let a and b be two given quantities. Again let
m1, m2, m3 ....... mn are n geometric means
between a and b. Then,
a, m1, m2, m3 ............., mn, b are in G. P.
Now, First term = a,
No. of means = n
Total no. of terms = n + 2
Last term (n + 2)th term = b
If r be the common ratio, then

9. ... r = (
𝑏
𝑎
)
1
𝑛+1
 m1 = ar
 m2 = a𝑟2
 m3 = a 𝑟3
.
.
.
 mn = a 𝑟 𝑛