• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer
  • ICSE Solutions
    • ICSE Solutions for Class 10
    • ICSE Solutions for Class 9
    • ICSE Solutions for Class 8
    • ICSE Solutions for Class 7
    • ICSE Solutions for Class 6
  • Selina Solutions
  • ML Aggarwal Solutions
  • ISC & ICSE Papers
    • ICSE Previous Year Question Papers Class 10
    • ISC Previous Year Question Papers
    • ICSE Specimen Paper 2021-2022 Class 10 Solved
    • ICSE Specimen Papers 2020 for Class 9
    • ISC Specimen Papers 2020 for Class 12
    • ISC Specimen Papers 2020 for Class 11
    • ICSE Time Table 2020 Class 10
    • ISC Time Table 2020 Class 12
  • Maths
    • Merit Batch

A Plus Topper

Improve your Grades

  • CBSE Sample Papers
  • HSSLive
    • HSSLive Plus Two
    • HSSLive Plus One
    • Kerala SSLC
  • Exams
  • NCERT Solutions for Class 10 Maths
  • NIOS
  • Chemistry
  • Physics
  • ICSE Books

What is Arithmetico–Geometric Sequence?

May 1, 2023 by Prasanna

What is Arithmetico–Geometric Sequence?

Arithmetico-geometric Progression (A.G.P.)

Definition:
The combination of arithmetic and geometric progression is called arithmetico-geometric progression.

nth term of A.G.P.

If a1, a2, a3, ……. an, ..…. is an A.P. and b1, b2, b3, ……. bn, ..…. is a G.P., then the sequence a1b1, a2b2, a3b3, …….., anbn, …….. is said to be an arithmetico-geometric sequence.
Thus, the general form of an arithmetico geometric sequence is a, (a + d)r, (a + 2d)r2, (a + 3d)r3, ………
From the symmetry we obtain that the nth term of this sequence is [a + (n – 1)d]rn–1.
Also, let a, (a + d)r, (a + 2d)r2, (a + 3d)r3, ……… be an arithmetico-geometric sequence.
Then, a, (a + d)r, (a + 2d)r2, (a + 3d)r3, ……… is an arithmetico-geometric series.

Sum of A.G.P.

What is Arithmetico–Geometric Sequence 1

Method for finding sum

This method is applicable for both sum of n terms and sum of infinite number of terms.
First suppose that sum of the series is S, then multiply it by common ratio of the G.P. and subtract. In this way, we shall get a G.P., whose sum can be easily obtained.

Method of difference

If the differences of the successive terms of a series are in A.P. or G.P., we can find nth term of the series by the following steps :
Step I: Denote the nth term by Tn and the sum of the series upto n terms by Sn.
Step II: Rewrite the given series with each term shifted by one place to the right.
Step III: By subtracting the later series from the former, find Tn.
Step IV: From Tn, Sn can be found by appropriate summation.

Example : Consider the series 1+ 3 + 6 + 10 + 15 +…..to n terms. Here differences between the successive terms are 3 – 1, 6 – 3, 10 – 6, 15 – 10, …….i.e., 2, 3, 4, 5,…… which are in A.P. This difference could be in G.P. also. Now let us find its sum:
What is Arithmetico–Geometric Sequence 2

Miscellaneous series

Special series

(1) Sum of first n natural numbers
What is Arithmetico–Geometric Sequence 3
(2) Sum of squares of first n natural numbers
What is Arithmetico–Geometric Sequence 4
(3) Sum of cubes of first n natural numbers
What is Arithmetico–Geometric Sequence 5

Filed Under: Mathematics Tagged With: Arithmetico-geometric Progression, Method for finding sum, Method of difference, Miscellaneous series, nth term of A.G.P.

Primary Sidebar

  • MCQ Questions
  • ICSE Solutions
  • Selina ICSE Solutions
  • Concise Mathematics Class 10 ICSE Solutions
  • Concise Physics Class 10 ICSE Solutions
  • Concise Chemistry Class 10 ICSE Solutions
  • Concise Biology Class 10 ICSE Solutions
  • Concise Mathematics Class 9 ICSE Solutions
  • Concise Physics Class 9 ICSE Solutions
  • Concise Chemistry Class 9 ICSE Solutions
  • Concise Biology Class 9 ICSE Solutions
  • ML Aggarwal Solutions
  • ML Aggarwal Class 10 Solutions
  • ML Aggarwal Class 9 Solutions
  • ML Aggarwal Class 8 Solutions
  • ML Aggarwal Class 7 Solutions
  • ML Aggarwal Class 6 Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC
  • Recent Posts

    • An Icon of Civil Rights Question and Answers
    • Freedom Question and Answers
    • A Long Walk to Freedom Question and Answers
    • The Ham Radio Question and Answers
    • Grabbing Everything on the Land Question and Answers
    • A Havoc of Flood Question and Answers
    • Can’t Climb Trees Any More Question and Answers
    • A Shipwrecked Sailor Summary
    • The River Question and Answers
    • What is Man without the Beasts? Question and Answers
    • Not Just a Teacher, but a Friend Question and Answers

    Footer

    • Picture Dictionary
    • English Speech
    • ICSE Solutions
    • Selina ICSE Solutions
    • ML Aggarwal Solutions
    • HSSLive Plus One
    • HSSLive Plus Two
    • Kerala SSLC
    • Distance Education
    DisclaimerPrivacy Policy
    Area Volume Calculator