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.
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...
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.
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!
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 $x,\,y,\,z,\,n$.
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...
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...
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...
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+...
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...
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.
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...
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...
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...
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...?
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.
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...
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...
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...
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}...
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 =...
Find three irreducible fractions $\dfrac{a}{d}$, $\dfrac{b}{d}$ and $\dfrac{c}{d}$ that form an arithmetic progression, if $\dfrac{b}{a}=\dfrac{1+a}{1+d}$, $\dfrac{c}{b}=\dfrac{1+b}{1+d}$.
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...
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...
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...
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
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...
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...
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...
"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...
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...
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?
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...
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...
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...
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
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)...
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...
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...
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...?
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.
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...
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 =...
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)...
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 ?
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...
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}...