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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 it is Arithmetic Progression then find the value of s?

The value of s = I started this question by making an A.P as the common difference is same and got the answer that I need number of terms to proceed further but my valie for number of terms is coming in fraction that is not possible i tried many…
Ankit Kumar
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Minimum number of terms in an arithmetic progression

I'm pretty sure I've heard this many times from people that I require at least $3$ terms to form an AP. But now, after the reading the definition of AP, An arithmetic progression(AP) or arithmetic sequence is a sequence of numbers such that the…
William
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Arithmetic progression (given third term and difference between 5th and 7th term)

Given that the 3rd term of an arithmetic progression (AP) is 16 and the difference between the 5th and the 7th term is 12, write down the first 7 terms of the AP. For an AP, the $n^{th}$ term is given by: $$a_n=a+(n-1)d$$ where $a$ is the…
user495760
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Why arithmetic progression formula $S_n = (a_1 + a_n)*n/2$ works with uneven number of integer members?

Let's consider arithmetic progression with integer numbers. Arithmetic progression sum $S_n = (a_1 + a_n)*n/2$, where $a_n=a_1+d(n-1) $ So $ S_n = (2*a_1 + d(n-1))*n/2 = a_1*n + d(n-1)*n/2$ I cannot understand, why it always happens that…
Code Complete
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Given the ratio of the sum of $n$ terms of two arithmetic progressions, find the ratio of their $m$-th terms

The ratio of sum of $n$ terms of two arithmetic progressions is $$7n + 1 : 4n +27$$ Find the ratio of their $m$-th terms. I tried everything but couldn't get the answer. Here is what I tried:- $\dfrac{\frac{n}{2}(2a+(n-1)d)}{\frac{n}{2}(2a…
Ram Keswani
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If $x$ be the $A.M$ between $y$ and$z$...

If $x$ be the AM between $y$ and $z$, $y$ be the GM between $z$ and $x$, then $x$, $y$, $z$ are in : $1$). A.P $2$). G.P $3$). H.P $4$). None. My Attempt: $x$ is the AM between $y$ and $z$ $$x=\dfrac {y+z}{2}$$ $$2x=y+z$$ $y$ is the G.M between $z$…
pi-π
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Prove that the coefficient of $n$ is the common difference.

Suppose the sequence <$a_n$> is an Arithmetic Progression if its $n^{th}$ term is a linear expression in $n$ then show that common difference is equal to the coefficient of $n$.
user401699
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Prove that $\frac{1}{b-c},\frac{1}{c-a},\frac{1}{a-b}$ are in arithmetic progression under given condition.

Suppose $(b-c)^2,(c-a)^2,(a-b)^2$ are in arithmetic progression. Then show that $\frac{1}{b-c},\frac{1}{c-a},\frac{1}{a-b}$ are also in an arithmetic progression. Please help.
John
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Finding the common difference in an arithmetic sequence for a special case

Suppose that $m^2S_m, mS_{m^2}, S_{m^3}$ are three arbitrary terms in an arithmetic sequence. These terms are also three successive terms in an geometric sequence. If in the arithmetic sequence, $S(20) = 20$, how can we find the common difference of…
Javad Kouhi
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Arithmetc Progression containing odd terms

Here is the question: If for an AP of odd number of terms,the sum of all the terms is $\frac{15}{8}$ times the sum of the terms in odd places then find the number of terms in the AP. my try:First of all i thought that in an odd AP there will be…
Kartik Watwani
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What is the least number of terms $a+nd$ required for a finite arithmetic progression?

I would like clarification on the following definition of finite arithmetic progression: According to Wikipedia, "A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic…
M. B. Jones
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From which term should this AP start?

Your friend Veer wants to participate in a 200 m race. He can currently run that distance in 51 seconds and with each day of practice it takes him 2 seconds less. He wants to do in 31 seconds. What is the minimum number of days he needs to practice…
Atul Anand
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Sum of terms whose differences are in Arithmetic progression.

My friend gave me a problem of finding sum of 10 terms. I simplified the terms and got: Terms: 3, 9, 18, 30, ... , 165 Differences: 6, 9, 12, ... , 30 I saw that the terms' differences are in AP. I got the sum 660 by simply adding them. Then I…
Arya
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Arithmetic progression and summation.

Had this question from some exam which goes like so: Let $V_r$ denote the sum of first $r$ terms of an arithmetic progression whose first term is $r$ and the common difference is $(2r-1)$. Let $T_r = V_{r+1} - V_{r}$ and $Q_r = T_{r+1}-T_r$ for…
Rew
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Relationship between arithmetic progression and inversely proportional rule of three.

I did a racing game, where you are a team manager. In this game, the cars complete laps on several circuits. I am having trouble reaching a definition in the lap time. Defining the speed of the car, I also define the time the car will complete the…
Boneco Sinforoso
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