Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Geometric Progression as the name suggests is a kind of sequence in which the terms increase geometrically. This simply means that every next element is obtained by multiplying the previous element by a constant. This constant which is obtained by dividing the successive term by the preceding term is termed as the common ratio. The common ratio is generally denoted by ‘r’ and is equal to
r = a_{2}/a_{1} = a_{3}/a_{2} = ………. = a_{n}/a_{n-1},
where r is the common ration and a_{i} denotes the ith term.
The geometric progression may also be termed as the geometric sequence or GP in short.
Now, we aim at finding the sum of first n terms of a geometric progression, generally abbreviated as G.P.
S_{n} = a + ar + ar^{2} +…+ ar^{n–1} … (1)
Multiplying both sides by r, we get,
rS_{n} = ar + ar^{2} +…+ ar^{n–1} + ar^{n} … (2)
Subtracting (7) from (6), we have
S_{n} – rS_{n} = a – ar^{n}
or, S_{n} = a(1–r^{n})/(1–r) … (3)
From equation (3), it follows that the sum of n terms of a G.P.
S_{n} = a(1–r^{n})/(1–r)
Hence, the sum of first n terms of a G.P. is given by the formula
S_{n} = a (1–r^{n})/ (1–r)
Watch this Video for more reference
We now wish to consider the case when n tends to infinity.
The case of n tending to infinity will depend on the value of |r|. we consider two cases:
If the value of |r| is greater than 1 then as n tends to infinity, r^{n }will also become infinite. Hence, in this case, S_{∞ }will also tend to infinity.
Similarly, if we consider the case when the value of |r| < 1, then as n tends to infinity, r^{n }tends to zero. In this case, S_{∞} = a/1–r for |r| < 1.
Result: We now discuss the cases when any real number is added, subtracted, multiplied or divided to each term of the geometric series.
1. If all the terms of a G.P. are multiplied or divided by a constant number, the resultant sequence is also a G.P.
Suppose a_{1}, a_{2}, a_{3}, ……, a_{n} are in G.P.
then ka_{1}, ka_{2}, ka_{3}, ……, ka_{n} and
a_{1}/k, a_{2}/k, ... ... ... a_{n}/k will also be in G.P.
where k ∈R and k ≠ 0.
2. The multiplication or division of two geometric progressions is also a geometric progression.
Suppose a_{1}, a_{2}, a_{3}, ……, a_{n} and b_{1}, b_{2}, b_{3}, ……, b_{n} are two G.P.
then a_{1}b_{1}, a_{2}b_{2}, a_{3}b_{3}, ……, a_{n}b_{n}
and a_{1}/b1, a_{2}/b_{2}, ... ... ..., a_{n}/b_{n} will also be in G.P.
3. Reversing the order of a G.P.’s also results a G.P.
then a_{n}, a_{n–1}, a_{n–2}, ……, a_{3}, a_{2}, a_{1} will also be in G.P.
4. Taking the inverse of a G.P. also results a G.P.
then 1/a_{1}, 1/a_{2}, 1/a_{3} ……, 1/a_{n} will also be in G.P.
Remark: Generally, questions are also asked in exam on finding the three numbers of G.P. satisfying certain condition. Though it is not mandatory, but if the numbers are assumed in a particular manner, it becomes easy to solve questions. We suggest students to assume the geometric variables in the following manner:
If we need to assume three numbers in G.P. then a/b, a, ab here common ratio is b
Four number in G.P. a/b^{3}, a/b, ab, ab^{3} here common ratio is b^{2}
Five numbers in G.P. a/b^{2}, a/b, a, ab, ab^{2} here common ratio is b
Illustration: Find the sum of the geometric series -2, ½, -1/8, …., -1/37268
Solution: Writing the terms of the geometric progression
Some key points to be noted:
If each term of a G.P. is multiplied or divided by a fixed non-zero constant then the new sequence is also a G.P. with the same common ratio.
If each term of a G.P. (with common ratio r) is raised to the power k, then the resulting sequence is also a G.P. with common ratio r^{k}.
If there are two G.P.s with different ratios say
a_{1}, a_{2}, a_{3}, ……, is a G.P. with common ratio r b_{1}, b_{2}, b_{3}, …… is the second G.P. with common ratio r’ respectively then the sequence a_{1}b_{1}, a_{2}b_{2}, a_{3}b_{3}, …… is also a G.P. with common ratio rr’.
We have stated above the process of assuming three, four or five terms in a G.P. In general, if we need to assume (2k + 1) terms in a G.P., they can be assumed as a/r^{k}, a/r^{k–1}, …, a, a^{r}, …, ar^{k}.
If 2k terms need to be assumed then the terms of the G.P. can be assumed as a/r^{2k–1}, a/r^{2k–3}, ... ... ... a/r, ar, ……, ar^{2k–1}.
If a_{1}, a_{2}, ……, a_{n} are in G.P., then a_{1}a_{n} = a_{2}a_{n–1} = a_{3}a_{n–2} = ……
If a_{1}, a_{2}, a_{3}, …… is a G.P. (each a_{1} > 0), then loga_{1}, loga_{2}, loga_{3} …… is an A.P. The converse is also true.
askIITians offers comprehensive study material which contains all the important topics of IIT JEE Mathematics. The most frequently asked questions of the JEE have also been covered in detail. It is important to master this topic in order to remain competitive in the JEE.
Click here for the Complete Syllabus of IIT JEE Mathematics.
Look into the Previous Year Papers with Solutions to get a hint of the kinds of questions asked in the exam.
You can get the knowledge of Useful Books of Mathematics here
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Harmonic Mean Table of Content Formula for...
IIT JEE Examples Based on Relationship between AM,...
IIT JEE Arithmetic Mean between two numbers | JEE...
Geometric Progression (G.P.) Table of contents...
Relationship between A.M. and G.M. Table of...
IIT JEE Arithmetic Progression | JEE Progression...
IIT JEE Sum of First N Terms Of AP | JEE The Sum...
IIT JEE Arithmetic Mean Examples | JEE Arithmetic...
Weighted Means Let a 1 , a 2 , ……, a...
IIT JEE Basic Principle of AP | JEE Principle of...
Solved Examples based on GP Solved examples :...
Sum to n terms of Special Series Table of contents...
Solved Examples Based on Harmonic mean...
Sequences Table of contents Meaning of Sequence...
Method of Differences Suppose a 1 , a 2 , a 3 ,...
Download IIT JEE Solved Examples Based on...
IIT JEE Basic Concepts of Sequences and Pattern |...
Relation between A.M., G.M. and H.M. Let there are...
Introduction of Sequences and Series Table of...
Harmonic Progression Harmonic progression is an...
Arithmeticogeometric Progression Solved Examples...
IIT JEE Arithmetic Geometric Progression | JEE...
IIT JEE Geometric Mean | JEE Geometric Mean of A...
Arithmetic Mean of mth Power The concept of...
Series Table of contents Meaning of Series History...