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Arithmetic, Geometric, Harmonic Progressions - With Problems and MCQ

Sum of first N Sixth Powers of Natural Numbers

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE 

Main or Advanced/AIEEE

, and anyone else who needs this Tutorial as a reference!

 (Tutorial on AP, GP, HP, AM, GM, HM and Series Summations)- With Solved Problems,  MCQ Quizzes

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on Progressions
MCQ #1: Arithmetic Progression 
MCQ #2: Geometric Progression
MCQ #3 : More on Geometric Progressions.
MCQ #4 : Harmonic Progressions. 
MCQ #5: More on Harmonic Progression
MCQ #6: Mixed Progressions

Quadratic Equations
MCQ Quadratic Equations

Quadratic In-equations
MCQ Quadratic In-equations
 Coordinate Geometry - Straight Lines
MCQ #1: Cartesian Planes, Straight Line Basics
MCQ #2 on Straight Lines
MCQ #3 on Straight Lines
MCQ #4 on Straight Lines

1 MCQ #1 on Circles. 
2 MCQ #2 on Circles. 
3 MCQ #3 on Circles. 

Conic Sections- Parabola, Hyperbola, Ellipse
1 MCQ- The Basics of Conic Sections
2 MCQ on Parabola..
3 MCQ on Hyperbola
4 MCQ on Ellipses. 
MCQ #1 on Basic Probability
MCQ #2: More Challenging Problems on Probability
MCQ #3- Conditional Probability and Bayes Theorem

Series and Progressions : AP, GP, HP 

After studying the chapter you might find it useful to attempt these Multiple Choice Question Quizzes to assess how well you understood the topic.

A sequence refers to a list of numbers with some inherent pattern. 
General term of a sequence, generally written as an is an expression that for different values of ‘n’ gives different terms of the sequence.

Arithmetic Progression, Terms and Summation

A sequence of the type a,a+d,a+2d,a+3d,… is said to form an AP.
Each term differs from the previous one by a fixed value called as “common difference”.
a and d are usually used to denote first term and common difference respectively. nth term of an AP is given by
The tutorial will show you how to compute the sum of the first n terms of an AP

Selecting terms in AP:

3 terms in an AP are – a-d,a,a+d
4 terms in an AP are – a-3d,a-d,a+d,a+3d
And so on.
mth term from the end of an AP : It is the (n-m+1)th term from the beginning.
Alternately, we can reverse the AP, and consider the first term and common difference of the new AP as last term and negative of common difference of the old AP.
If a constant is added, or subtracted, or multiplied or divided from every term on an AP, the resulting sequence is still an AP. 

Arithmetic Mean: 

if a,b,c are in AP, then b is said to be arithmetic mean of a and c.
Equivalently, if 2b=a+c, then a,b,c are in AP. If between a and b we are to insert n numbers A1,A2,A3,….,An such that a, A1,A2,A3,….,An form an AP, then A1,A2,A3,….,An are called n Arithmetic Means.

Geometric Progression, Terms and Summation

A sequence of the type: a,ar,ar2,ar3,… is called a GP.
In a GP, the nth term of a GP is given by: an=arm-1
The tutorial will show you how to compute the sum of first n terms of a GP.
When r=1, the GP can be treated as an AP, and clearly the sum of first n terms then is “nXa”.
Similar to  an AP, mth term from the end is (n-m+1)th term from the starting.

Selecting terms in a GP:

A GP of 3 terms : a/r,a,ar
A GP of 4 terms:a/r3,a/r,ar,ar3
And so on.
If all terms in a GP are multiplied or divided by the same number, or are raised to the same power, then the resulting sequence is still a GP.

Geometric Mean:

If a,b,c are in GP then b2=ac and b is called the GM of a and c. Conversely, if b2=ac, then a,b,c are in GP.
Sum of infinite terms of a GP: If -1<r<1, then GP is said to converge, that is to say that sum of infinite terms of such a GP tends to a constant value. 

Harmonic Progression

If a sequence is in AP, then the sequence obtained by taking the reciprocal of every term in the sequence forms an HP.
That is if a,b,c,… form an AP, then 1/a,1/b,1/c,… form an HP.
Let a,b,c form an HP. Then clearly, 1/a,1/b,1/c form an AP. 
b is called the Harmonic Mean.
Problems related to HP are generally solved by converting it into an AP.

Relationship between Arithmetic, Harmonic and Geometric Means

NOTE: Let a,b be two positive real numbers. Then
Also, AM>GM>HM

Here are some of the problems solved in the tutorial : 

Q:  What is the common difference of an AP whose nth term is xn+y?
Q:  If the sum of n terms of an AP is 2n2+3n, what is the kth term?
Q:  If am=n and an=m, what is ap?
Q:  If the sums of n terms of two arithmetic progressions are in the ratio 2n+5:3n+4, then what is the
ratio of their nth terms?
Q:  If sum of n terms of an AP is 3n2+5n then which of its terms is 164?
Q:  If sum of first p terms is q and sum of first q terms is p then what is the sum of first p+q terms ?
Q:  If four numbers are in AP such that their sum is 50, and the greatest number is four times the least,
then what are the numbers?
Q:  If n arithmetic means are inserted between 1 and 31 such that ratio of first and nth mean is 3:29,
then what is the value of n?
Q:  If S1 is the sum of an AP of ‘n’ odd number of terms and S2 the sum of terms of the series in odd
places, then what is S1/S2?
Q:  What is the sum of 2 digit multiples of 4?
Q:  Find the sum of 2+6+18+…+4374.
Q:  Find the sum of 5+55+555+…n terms
Q:  Find the sum of n terms of the sequence whose nth term is 2n+3n
Q:  Find a GP for which the sum of first two terms is -4 and the fifth term is 4 times the third term.
Q:  Find the least value of n for which the sum of 1+3+9+27+… to n terms is greater than 7000.
Q:   If a,b,c are in GP, prove that log a, log b, log c are in AP and vice versa.
Q:  If a,b,c are in GP and x,y are AM’s between a,b and b,c respectively, then prove that x,b,y are in HP.
Q:  Find the sum to n terms of series 1.2.3+2.3.4+3.4.5+4.5.6+…
Q:  If a, b, c are in H.P., show that a/b+c, b/c+a, c/a+b are also in H.P.
Q:   Evaluate: 62 + 72 + 82 + 92 + 102
Q:  Prove that (a + b + c) (ab + bc + ca) > 9abc.
Q:  If a2, b2, c2 are in A.P. show that b+ c, c, c+ a, a + b are in H.P.
Q:  If the AM between ‘a’ and ‘b’ is twice as greater as the GM, show that a/b = 7+4√3
Q:  Sum to the n terms the series, 12-22+32-42+… n is even.

Standard Sums of first N terms of common sequences

The Complete Tutorial with Solved Problems (MCQ Quizzes/Worksheets after this)

MCQ Quiz



Arithmetic Progression 

MCQ Quiz: Arithmetic Progression

MCQ Quiz/Worksheet #2

Geometric Progression

MCQ Quiz: Geometric Progression

 MCQ Quiz/Worksheet #3

More on Geometric Progressions. 

Fill up the answers in the Answer Submission Form below this Question-Paper document. Your score will be emailed to you.

Questions: MCQ Quiz for Geometric Progressions

Answer Submission Form for MCQ Quiz #3

Answer Submission Form: Geometric Progression MCQ

MCQ Quiz/Worksheet #4 

Harmonic Progressions. 

Fill up the answers in the Answer Submission Form below this Question-Paper document. Your score will be emailed to you.

Questions: Harmonic Progression

Answer Submission Form for MCQ Quiz #4

Answer Submission Form: Harmonic Progression

MCQ Quiz/Worksheet #5

More on Harmonic Progression

MCQ Quiz: Harmonic Progression

MCQ Quiz/Worksheet #6

Mixed Progressions

MCQ Quiz: Harmonic Progression

You might like to take a look at our other algebra tutorials:

 Introduction to Complex Numbers
Introduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics.
 The Principle of Mathematical Induction Introductory problems related to Mathematical Induction.Quadratic Equations
Introducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation.  Cubic and higher order equations - relationship between roots and coefficients for these. Graphs and plots of quadratic equations.
Quadratic Inequalities
 Quadratic inequalities. Using factorization and visualization based methods.
 Series and Progressions
Arithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.