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Questions tagged [arithmetic-progressions]

Questions related to arithmetic progressions, which are sequences of numbers such that the difference between consecutive terms is constant

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the common difference between consecutive terms is constant. For instance, the sequence 15, 13, 11, 9, 7, $\ldots$ is an arithmetic progression with common difference –2.

If the first term of an arithmetic progression is $a_1$, and the common difference is $d$, then the $n$th term of the sequence $(a_n)$ is $$a_n = a_1 + (n-1)d.$$

If the common difference $d$ is—

  • positive, the terms increase to positive infinity.
  • negative, the terms decrease to negative infinity.

A finite portion of an arithmetic progression is called a finite arithmetic progression or sometimes just an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

1022 questions
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If p,q,r are positive and are in A.P the roots of quadratic equations $px^2+qx+r=0$ are real for

The answer is $|\frac rp -7|\ge 4\sqrt 3$ Since they are in AP $$2q=p+r$$ For x to be real $$q^2-4pr\ge 0$$ Then $$(\frac{p+r}{2})^2-4pr\ge 0$$ $$p^2+r^2-14pr\ge 0$$ I don’t know that to do next. Please help. Thanks!
Aditya
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Finding an array index mathematically [noob]

I won't get much out of pure LaTeX just to say this. But my question is: if I have an array with 2 values in it, sorted as x[0-255],y[0-255] where I have every possible combination, 65536 How can I do this in algebra? I'm writing a computer program…
thexiv
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Help with how to show aritmetic progression question.

How can I show that if $(\chi_{n})$ is a aritmetic progression, then: $$\frac{1}{ \sqrt{\chi_{1}} + \sqrt{\chi_{2}} } + \frac{1}{ \sqrt{\chi_{2}} + \sqrt{\chi_{3}} } + \cdots + \frac{1}{ \sqrt{\chi_{n-1}} + \sqrt{\chi_{n}} } = \frac{n-1}{…
Daniel
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If $a_1,a_2,....,a_n$ are in A.P, then prove that:

We have to prove that $\frac{1}{a_1a_n} +\frac{1}{a_2a_{n-1}}+....+\frac{1}{a_{n-1}a_2} +\frac{1}{a_aa_1} = \frac{2}{a_1+a_n}(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n})$ The middle term will depend on the fact that $n$ is even or odd. So, I…
tomriddle99
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Find the general term of the sequence.

The given sequence is : $2\frac{1}{2}, 1\frac{7}{13}, 1\frac{1}{9}, \frac{20}{23}, ........$. I subtracted the 2nd and 1st term and the result was $-\frac{10}{13}$ and then the 3rd and 2nd term and the result was $-\frac{50}{117}$.
tomriddle99
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Ratio of two different AP series

Consider $A_1,A_2,A_3,.....A_n$ and$B_1,B_2,B_3,.....B_n$ $\ge20$ are two different Arithmetic progression such that $\frac{A_n}{B_1}=\frac{\sum_{i=1}^{n}2A_i}{\sum_{i=1}^{n}B_i}=\frac{B_n}{A_1}=4$, then find the value of i)…
Samar Imam Zaidi
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If $x,y,z$ are three natural numbers are in A.P. and $x+y+z=21$ then the possible number of values of the ordered triplet $(x,y,z)$ is

I have assumed $x=a-d,y=a$ and $z=a+d$ by which I get $a=7$ and numbers could be $1,7,13$ or $13,7,1$ or $7,7,7$, but the total no of solutions set is $13$. I am not getting it. Please help.
Faaiz Ali
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A question about AP.

How is the encircled step justifiable? According to my knowledge I can substitute m=any variable but how can I substitute m=2m-1, isn't it the same as assuming m=1?
Raknos13
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It's based on Arithmetic Progression, on finding the sum of an AP.

Find the $S_{15}$ if $T_3 = 18$ and $T_{10} = 67$. Find the sum in this AP.
Alex Jones
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If $pth$ term of an A.P is $q$ and its $qth$ is $ p$, show that its $rth$ term is $p+q-r$. What is its $(p+q)th $term?

p$th$,q$th$ and r$th$ term are $q, p, p+q-r$ respectively. $a_{p}= a+(p-1)d = q$ $= a+pd-d=q$ $a_{q} = a+(q-1)d = p$ $=a+qd-d=p$ Now for getting common difference we should minus 2nd term from first term that means p$th$ term minus…
bappy
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Finding the maximum sum of a decreasing Arithmetic Progression

Find the maximum sum of the AP $40+38+36+34+32+...$ My Attempt: $a=40$ $d=(-2)$ $S_n= \frac{n}{2}[2a+(n-1)d]$ $S_n= \frac{n}{2}[80+(n-1)(-2)]$ $S_n= n[40+(n-1)(-1)]$ $S_n= n[41-n]$ $S_n= 41n-n^2$ I can find the answer by differentiating…
oshhh
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How to find the sum of the first 21 terms of an Arithmetic Progression(A.P)?

Question. If the sum of first $12$ terms of an A.P. is equal to to the sum of the first $18$ terms of the same A.P., find the sum of the first $21$ terms of the same A.P. $a=$ first term $d=$ difference I know now, $2(2a + 11d) = 3(2a +…
mac07
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arithmetic progession question

if $\sqrt{a-x}, \sqrt x, \sqrt{a+x}$ are in AP provided $a>x$ and $a,x$ are positive integers then what is the least possible value of $x$?
user319310
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How to test if arithmetic progression is subset of another?

I have two arithmetic progressions $R = \{ a + rn, n \ge 0 \}$ and $S = \{ b + sn, n \ge 0 \}$. What are necessary and sufficient conditions to test whether R is a subset of S?
Colonel Panic
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ratio of arithmetic summation

This is the question which I am referring to Find the AP in which the ratio of the sum to n terms to the sum of succeeding n terms is independent of n. What I have thought: we are talking about the following ratio…
Kartik Watwani
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