What is Arithmetic progression: Definition and 56 Discussions

An Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.
If the initial term of an arithmetic progression is



{\displaystyle a_{1}}
and the common difference of successive members is d, then the nth term of the sequence (



{\displaystyle a_{n}}
) is given by:








{\displaystyle \ a_{n}=a_{1}+(n-1)d}
,and in general








{\displaystyle \ a_{n}=a_{m}+(n-m)d}
.A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

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  1. brotherbobby

    Prove that terms cyclic in ##a,b,c##, are in ##\text{AP}##

    Problem statement : Let me copy and paste the problem as it appears in the text to the right. Attempt : We have the terms of the ##\text{AP}## as ##a, \;b = a+d, \;c = a+2d## Let the first term of the required expression be ##t_1 = a^2(b+c) = a^2(2a+3d)=2a^3+3a^2d\dots\quad (1)## Let the second...
  2. brotherbobby

    Ratio of the sum of ##n## terms of two AP series

    Statement of the problem : I copy and paste the problem as it appears in the text to the right. Attempt : I must admit I didn't get far, but below is what I did. I use ##\text{MathType}^{\circledR}## hoping am not violating anything. Request : A hint would be very welcome.
  3. pairofstrings

    B Arithmetic progression, Geometric progression and Harmonic progression

    How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!
  4. anemone

    MHB Arithmetic Progression: Expressing d in Terms of x,y,z,n

    Let $a_1,a_2,\,\cdots,\,a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let (1) $a_1^2+a_3^2+\cdots+a_{2n-1}^2=x$ (2) $a_2^2+a_4^2+\cdots+a_{2n}^2=y$ (3) $a_n+a_{n+1}=z$ Express $d$ in terms of $x,\,y,\,z,\,n$.
  5. AN630078

    Solving for nC8,nC9, and nC10 in an Arithmetic Progression

    Well, I am having a little difficulty knowing how to approach finding a solution to this problem. I am aware that in an arithmetic progression the first term is a and there is a constant common difference defined as d=un+1-un Expanding the binomial given...
  6. Purpleshinyrock

    Arithmetic sequences Problem

    Summary:: Sequences, Progressions Hello. I have been Given the following exercise, Let (a1, a2, ... an, ..., a2n) be an arithmetic progression such that the sum of the last n terms is equal to three times the sum of the first n terms. Determine the sum of the first 10 terms as a function of...
  7. WMDhamnekar

    MHB Find Common Difference of A.P. Given G.P. & Logarithms

    If a,b, c, are in G.P and $\log_ba, \log_cb,\log_ac$ are in A.P. I want to find the common difference of A.P. Answer: After doing some computations, I stuck here. $\frac{2(\log a+\log r)}{\log a+2\log r}=\frac{2(\log a)^2+3\log r\log a +2(\log r)^2}{(\log a)^2+\log r\log a}$ How to proceed...
  8. S

    Solving for the Sum of an Arithmetic Progression | m>n | AP Homework

    Homework Statement In an AP, sum of first n terms is equal to m and sum of first m terms is equal to n. Then, find the sum of first (m-n) terms in terms of m and n, assuming m>n. Homework Equations Sum of an AP: n/2 * {2a+ (n-1)d} The Attempt at a Solution We get two equations: m= n/2 * {2a+...
  9. donaldparida

    Find α+β+αβ: 7 "Solving for α+β+αβ in Arithmetic Progression

    Homework Statement P(x) =ax2+bx+c where a, b and c are in arithmetic progression and are positive. α and β are the roots of the equation and are integers. Find the value of α+β+αβ. (Answer is 7) Homework Equations x = {−b ± √(b2 − 4ac)} /2a 3. The Attempt at a Solution [/B] Since a, b and c...
  10. M

    MHB Find Lowest Value for A: a1, a2, a3 & 4 | Arithmetic Progression

    a1, a2, a3 and 4 make an arithmetic progression with difference d. For which values of d, A = a1a2 + a2a3 + a3a1 has the lowest value?I don't know if I went with the right approach, but I managed to get this : A=3x2 +6xd + 2d2 for a1= x, a2 = x + d, etc... But I don't know what else to do.
  11. K

    Arithmetic progression

    Homework Statement A finite arithmetic progression is given such that ##S_n>0## and ##d>0##. If the first member of the progression remains the same but ##d## increases by 2, then ##S_n## increases 3 times. If the first member of the progression remains the same but ##d## increases 4 times...

    Solving Arithmetic Progression: Sum and Product of Four Integers

    Homework Statement the sum of four integers in A.P is 24 and their product is 945.find themHomework Equations ##(a-d)+a+(a+d)+(a+2*d)=24## ##2a+d=12## ##(a+d)(a-d)(a)(a+2d)=945## ##(a^2-d^2)(a^2+2*a*d)=945## The Attempt at a Solution there are two equations and two unknowns a(one of the...
  13. T

    Arithmetic progression. find p.

    Homework Statement johns father gave him a loan of $1080 to buy a car. the loan was to repaid in 12 monthly installments starting with an intial payment of $p in the 1st month. there is no interest charged on the loan but the installments increase by $60/month. a) show that p = 570 and find in...
  14. osirvics

    Arithmetic Progression: Find Common Difference with Given First Term and Ratio

    Homework Statement The first term of an a.p is -8, the ratio of the 7th term to the 9th term is 5:8. what is the common difference of the progression? Homework EquationsThe Attempt at a Solution I've tried... it confuses me. Can anyone give me some hints or tips...?
  15. C

    MHB Arithmetic Progression: Finding the First Term and Common Difference

    The sum of the first 100 terms of an arithmetic progression is 15050; the first, third and eleventh terms of this progression are three consecutive terms of a geometric progression. Find the first term, a and the non-zero common difference, d, of the arithmetic progression.
  16. Suraj M

    Arithmetic progression sum and nth term

    Homework Statement The ratio of sums of 2 AP for n terms each is ## \frac{3n + 8}{7n + 15}## that is $$ {\frac{s_a}{s_b}} = \frac{3n + 8}{7n + 15} $$ find the ratio of their 12th terms. $$ Required= \frac{a₁_a+(n-1)d_a}{a_b + (n-1)d_b}$$Homework Equations Tn = a + (n-1)dThe Attempt at a...
  17. anemone

    MHB Roots of $g'(x)$ in AP: Proving the Theory

    The roots of a fourth degree polynomial $g(x)=0$ are in an AP (arithmetic progression). Prove that the roots of $g'(x)=0$ must also form an AP.
  18. A

    Binomial series with coeficients in arithmetic progression

    Homework Statement The binomial expansion of (1+x)^n, n is a positive integer, may be written in the form (1+x)^{n} = 1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+...c_{r}x^{r}+... Show that , if c_{s-1}, c_{s} and c_{s+1} are in arithmetic progression then (n-2s)^{2} =n+2 Homework Equations The Attempt...
  19. A

    Problem about arithmetic progression

    Hi, can't solve following prob: Let a, b and c be real numbers. Given that a^2, b^2 and c^2 are in arithmetic progression show that 1 / (b + c), 1 / (c + a) and 1 / (a + b) are also in arithmetic progression. From assumptions: b^2 = a^2 + nk and c^2 = b^2 + mk where k is some real number...
  20. U

    Arithmetic Progression problem

    Homework Statement Let a1,a2,a3...,a4001 are in A.P. such that \dfrac{1}{a_1a_2}+\dfrac{1}{a_2a_3}+.......\dfrac{1}{a_{4000}a_{4001}} = 10 and a2+a4000=50. Then |a1-a4001| The Attempt at a Solution \dfrac{1}{a_2} \left( \dfrac{1}{a_1} + \dfrac{1}{a_3} \right) + \dfrac{1}{a_4}...
  21. z.js

    Arithmetic Progression Problem

    Homework Statement If the sum of the first 7 terms of an arithmetic progression is 28 and the sum of the first 15 terms is 90, find the sum of n terms.:eek: Homework Equations Sn = 0.5n[2a+(n-1)d] a is the first term and d is the common difference. n is the number of terms. nth term =...
  22. anemone

    MHB Arithmetic Progression Problem

    Find three irreducible fractions $\dfrac{a}{d}$, $\dfrac{b}{d}$ and $\dfrac{c}{d}$ that form an arithmetic progression, if $\dfrac{b}{a}=\dfrac{1+a}{1+d}$, $\dfrac{c}{b}=\dfrac{1+b}{1+d}$.
  23. P

    MHB Arithmetic progression question

    Hey, What is the greatest number a k-term arithmetic progression starting with 1 can end in if each term is less than or equal to n? I'm looking to write this as an expression involving n and k in order to count the number of arithmetic progressions of length k with each term in $[n]$, that is...
  24. anemone

    MHB Arithmetic Progression Challenge

    Find distinct positive integers a,\;b, and c such that a+b+c,\;ab+bc+ac,\;abc forms an arithmetic progression.
  25. H

    Arithmetic progression used to determine geometric progression

    Homework Statement an arithmetic progression(a1-a9) has 9 numbers. a1 equals 1 The combination(S) of all of the numbers of the arithmetic progression is 369 a geometric progression(b1-b9) also has 9 numbers. b1 equals a1(1) b9 equals a9(unknown) find b7 Homework Equations...
  26. ArcanaNoir

    Arithmetic progression topology, Z not compact

    Homework Statement The Dirichlet Prime Number Theorem indicates that if a and b are relatively prime, then the arithmetic progression A_{a,b} = \{ ...,a−2b,a−b,a,a+b,a+2b,...\} contains infinitely many prime numbers. Use this result to prove that Z in the arithmetic progression topology is not...
  27. N

    MHB Geometric Progression sequence with an Arithmetic Progression grouping problem

    Good Day, My friends and I are stuck on solving the last part of the attached problem. The solution is 2^[(n^2 + n)/2] - 1. Can anyone help us with solving this? Thanks & Regards, Nicodemus
  28. Government$

    Another arithmetic progression problem

    Homework Statement Sum of first three members of increasing arithmetic progression is 30 and sum of their squares is 692. What is the sum of the first 15 members?The Attempt at a Solution So i have system of equations: a1 + a2 + a3 = 30 (a1)^2 + (a2^2) + (a3^2) = 692...
  29. Government$

    Arithmetic progression problem

    Homework Statement Let a_{m+n}=A and a_{m-n}=B be members of arithmetic progression then a_{m} and a_{n} are? (m>n).The Attempt at a Solution I fugured that a_{m}=\frac{A+B}{2} but i have no idea what a_{n} is. In my textbook solution is a_{n}=\frac{(2n-m)A + mB}{2} How did they arrived to...
  30. trollcast

    Arithmetic Progression - show that question

    Homework Statement Given that a2, b2 and c 2 are in arithmetic progression show that: $$\frac{1}{b+c} , \frac{1}{c+a} , \frac{1}{a+b} $$ ,are also in arthimetic progression. Homework Equations The Attempt at a Solution So I assume by "in arithmetic progression" they mean those...
  31. FeDeX_LaTeX

    Prime Number Arithmetic Progression

    "Determine the least possible value of the largest term in an arithmetic progression of seven distinct primes." I really have no clue what to do here. Is there a general tactic that you can use to do this, other than trial and error? Some experimenting gives you these of arithmetic...
  32. B

    Interesting problem involving arithmetic progression

    I just came up with a problem I hope you will find interesting, but I can't seem it solve it myself. I thought of induction as some guide, but am not sure how to proceed. There are N terms in some finite arithmetic progression. Two of those terms are equal to 3. Prove that all terms in this...
  33. K

    Arithmetic Progression formula proof

    The proof says that - Let, Sn= a+(a+d)+(a+2d)+...+(a+(n-2)d)+(a+(n-1)d)----->1 Sn= (a+(n-2)d)+(a+(n-1)d)+...+a+(a+d)+(a+2d)------>2 Now if we have to add such things(1 and 2) how would we do that?
  34. C

    Particle Motion: Retardation & Arithmetic Progression

    Homework Statement A particle moves in a straight line away from a fixed point O in the line, such that when its distance from O is x its speed v is given by v=k/x , for some constant k. (a) show that the particle has a retardation which is inversely proportional to x3 The answer is...
  35. J

    Erdos conjecture on arithmetic progression

    I read this through wikipedia and some other sources and find it to be unsolved. Erdos offer a prize of $5000 to prove it. A mathematician at UW has looked at it and verify them to be correct. However, i still have some doubt about it because the proof i give is pretty simple. Can anyone take a...
  36. Saitama

    Arithmetic Progression question

    Homework Statement If x ε R, the numbers 51+x+51-x, a/2, 25x +25-x form an AP, then 'a' must lie in the interval:- a)[1,5) b)[2,5] c)[5,12] d)[12,∞) Homework Equations Not required The Attempt at a Solution I substituted y=5x. The terms are in AP, so the common difference is...
  37. N

    Solving Arithmetic Progressions

    Homework Statement The sum of the first 8 terms of an AP is 56, and the 6th term is 4 times the sum of the 2nd and the 3rd. Find the first term and the common difference Homework Equations The Attempt at a Solution 8th term = 56 6th term = 4x2nd+3rd
  38. V

    Summing up an Arithmetic Progression via Integration?

    Why doesn't the integration of the general term of an A.P. give its sum? Integration sums up finctions, so if I integrate the general term function of an A.P., I should get its sum. Like 2,4,6,8,... T=2+(n-1)2=2n \int T dn=n^2 ..(1)...
  39. S

    Arithmetic progression

    Homework Statement A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce increased by 3 people. How many people were there in the workforce after 26 weeks? Each member of the workforce earned $600 per week. What was the total wage bill...
  40. M

    Arithmetic progression question

    Homework Statement Series Q is an arithmetic series such that the sum of its first n even terms is more than the sum of its first n odd terms by 4n. Find the common difference of the series Q. The answer provided is 4. Homework Equations The Attempt at a Solution I have no ideas on this...
  41. M

    Arithmetic progression homework

    Homework Statement Need help with number (9).. Homework Equations The Attempt at a Solution Can anyone give me some hints? Thanks.
  42. M

    Sum of Positive Integers Less Than 150 Not Multiples of 5 or 7

    Homework Statement Find the sum of the positive integers which are less than 150 and are not multiples of 5 or 7. Homework Equations The Attempt at a Solution I tried it... Can anyone give me some hints or tips...?
  43. T

    What are the three terms in an A.P. with a sum of 36 and a product of 1428?

    This isn't a homework question, it's in a textbook I have and I'm a bit stumped. I know there's something relatively simple I'm missing so any help would be much appreciated (working too). Three consecutive terms of an A.P. have a sum of 36 and a product of 1428. Find the three terms.
  44. S

    Arithmetic progression of prime numbers

    what is the maximum number of terms can a arithmetic progression of only prime numbers have?
  45. H

    Arithmetic Progression: Finding the Sum of Terms with Given Conditions

    Homework Statement the first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200. Find the sum of all the terms in the progression Homework Equations The Attempt at a Solution I want to ask : what is the...
  46. M

    How many hours does it take to fill a 16m x 7m x 7m tank with water?

    Homework Statement water fills a tank at a rate of 150 litres during the first hour, 350 litres during the second hour, 550 litres during the 3rd hour and so on. find the number of hours neccesary to fill a rectangular tank 16m x 7m x7mHomework Equations l=a+(n-1)d S= n/2 (a+l) where: l =...
  47. E

    Mathematical induction and arithmetic progression

    Homework Statement All the terms of the arithmetic progression u1,u2,u3...,un are positive. Use mathematical induction to prove that, for n>= 2, n is an element of all positive integers, [ 1/ (u1u2) ] + [ 1/ (u2u3) ] + [ 1/ (u3u4) ] + ... + [ 1/ (un-1un) ] = ( n - 1 ) / ( u1un)...
  48. P

    Verifying the Sum of an Arithmetic Progression: Is it -(p+q) or +(p+q)?

    my book says that if sum of p terms of an ARTHMETRIC PROGRESSION is q and sum of q terms is p , then sum of p+q terms will be -(p+q) , but i am getting it as +(p+q), can someone verify it ?
  49. P

    Arithmetic Progression + System of equations + binomial

    Homework Statement A third degree polynomial has 3 roots that, when arranged in ascending order, form an arithmetic progression in which the sum of the 3 roots equal 9/5. The difference between the square of the greatest root and the smallest root is 24/5 Given that the coefficient of the...
  50. F

    Arithmetic Progression

    Homework Statement An arithmetic progression has n terms and a common difference of d. Prove that the difference between the sum of the last k terms and the sum of the first k terms is | (n-k)kd |. Homework Equations \begin{array}{l} {S_n} = \frac{n}{2}\left[ {2{a_1} + \left( {n - 1}...