Geometric Progression

Raj Kumar
06/09/2018 0 0

If a is the first term and r  is the common ratio of the geometric progression,

then its nth term is given by an = arn-1 

The sum Sn of the first n terms of the G.P. is given by

Sn = a (rn – 1)/ (r-1), when r ≠1; = na  if r =1             

If -1 < x < 1, then limxn = 0, as n →∞. Hence, the sum of an infinite G.P. is 1+x+x2+ ….. = 1/(1-x) 

If -1 < r< 1, then the sum of the infinite G.P. is a +ar+ ar2+ ….. = a/(1-r)

If each term of the G.P is multiplied or divided by a non-zero fixed constant, the resulting sequence is again a G.P.

 

If a1, a2, a3, …. andb1, b2, b3, … are two geometric progressions, then a1b1, a2b2, a3b3, …… is also a geometric progression and a1/b1, a2/b2, ... ... ..., an/bn will also be in G.P.  

Suppose a1, a2, a3, ……,an are in G.P. then an, an–1, an–2, ……, a3, a2, a1 will also be in G.P.

Taking the inverse of a G.P. also results a G.P. Suppose a1, a2, a3, ……,an are in G.P then 1/a1, 1/a2, 1/a3 ……, 1/an will also be in G.P   

If we need to assume three numbers in G.P. then they should be assumed as   a/b, a, ab  (here common ratio is b)

Four numbers in G.P. should be assumed as a/b3, a/b, ab, ab3 (here common ratio is b2)

Five numbers in G.P. a/b2, a/b, a, ab, ab2  (here common ratio is b)    

If a1, a2, a3,… ,an is a G.P (ai> 0 ∀i), then log a1, log a2, log a3, ……, log an is an A.P. In this case, the converse of the statement also holds good.

If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P., then b

 = √ac is the geometric mean of a and c.

0 Dislike
Follow 2

Please Enter a comment

Submit

Other Lessons for You

Geometric Progression : Basics
Geometric Mean (GM) The geometric mean is similar to the arithmetic mean, but it uses the property of multiplication, to find the mean between any two numbers. GM of two numbers a and b may be defined...

Hemant P. | 30/05/2018

0 0
0

copycat syndrome
To be good at something, it is better to copy. Most projects are now on google and reports are just copy pasted into a book. Duplication is a mantra in most marketing firms and some copy to the T. Effective...

Suraj Mohan | 22/10/2016

0 0
0

internal and external divison of vectors
If A and B are two points with position vectors a and b respectively and C is a point which divides AB internally in the ratio m: n, then the position vector of C is given by OC = (mb + na)/ (m + n). This...

Raj Kumar | 11/09/2018

0 0
0

Vectors and scaler
Vector is that quantity that has direction and magnitude for example force, velocity, acceleration and displacement The scaler is that quantity which has the only magnitude for example speed, volume,...

Raj Kumar | 11/09/2018

0 0
0

Notes on Arithmetic progression
If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d The sum, Sn of the first ‘n’...

Raj Kumar | 23/08/2018

0 0
0

Best Tutors and Trainers in your Locality Find Now »

UrbanPro.com is India's largest network of most trusted tutors and institutes. Over 25 lakh students rely on UrbanPro.com, to fulfill their learning requirements across 1,000+ categories. Using UrbanPro.com, parents, and students can compare multiple Tutors and Institutes and choose the one that best suits their requirements. More than 6.5 lakh verified Tutors and Institutes are helping millions of students every day and growing their tutoring business on UrbanPro.com. Whether you are looking for a tutor to learn mathematics, a German language trainer to brush up your German language skills or an institute to upgrade your IT skills, we have got the best selection of Tutors and Training Institutes for you. Read more