# 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

a

1

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

a

n

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

a

n

=

a

1

+
(
n

1
)
d

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

a

n

=

a

m

+
(
n

m
)
d

{\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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...
12. ### 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...

36. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### Arithmetic progression of prime numbers

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