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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.

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Arithmetic Progression generation with sum of all elements , first and last elements provided

Is there a method to find the common difference between elements of a progression with the first element, last element and the sum of all elements provided? The sum of all elements of the progression must match the value provided.
Sandeep Sugathan
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A question regarding the common terms of two APs.

What is the sum of first $50$ terms common to the $AP$ $15,19,23,\dots$ and the $AP$ $14,19,24,\dots$? I know that: The common terms start from $19$ and nothing else. I have tried this but I am facing a lot of difficulty. Please help me.
Yami Kanashi
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Infinitely many squares in any AP with integer $a, d$.

Following from questions 2090325 and 2090607, is it possible to show that any AP with first term $a$ and common difference $d$, where $a,d\in\mathbb N$ must contain a square and, following from question 2090325, infinitely many squares? If not, then…
Hypergeometricx
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Find the sum of $n$ term of the given AP $0.9, 0.91, 0.92, 0.93...............$?

First term $a= 0.9$, Common difference $d=0.01$ $$S_{n} = \frac n 2[2a+(n-1)d]$$ $$=\frac n 2[2(0.9)+(n-1)(0.01)]$$ $$=n(n+179)/200$$ Now my question is how we get $=n(n+179)/200$. Can explain how we get this answer. Thank you.
changer
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Arithmetic Progression (AP) - In the question, it will entail no sequence numbers, it will be based on two terms. The 8th Term and the 16th Term

If the 8th term of an AP is 36 and the 16th term is 68 Find: a) the first term. b) the common difference. c) The 20th term.
Bex John
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Arithmetic problem with logarithm

This is the question which I am referring to if log(base 3 of 2), log (base 3 of (2^(x)-5)), log(base 3 of (2^(x)-(7/2)) are in AP, find the value of x My try and answer I used the equation 2b= a+c , where a, b, c are in AP and solved for x so I…
Kartik Watwani
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If $ \frac{1}{a}, \frac {1}{b} , \frac{1}{c}$ are consecutive terms of an arithmetic sequence, express b in terms of a and c.

If $ \frac{1}{a}, \frac {1}{b} , \frac{1}{c}$ are consecutive terms of an arithmetic sequence, where $a,b,c \in \mathbb R\setminus 0$. Express $b$ in terms of $a$ and $c$. Answer in the most simplest form.
Darknight798
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Find x in a sequence of Arithmetic Progression (AP)

If $(8x+1), (6x-1)$ and $(3x+5)$ are in an AP, find the value of $x$. This sum is from a question paper and there is no other information given. I was able to solve other sums which had more information available. But I'm stuck at this sum.
NAPOLEON039
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Arithmetic Sequence Q11. Maths textbook standard level

For an arithmetic sequence $U_n$, $U_5 + 2U_3 = U_{12}$. If $U_7 = 25$, find an expression for the general term, $U_n$.
Erin Agatha
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How can I get the first element in this sequence?

There is an arithmetical sequence with 3n number of elements the sum of the last 2n elements is twice as big as the sum of the first 2n elements. And the sum of the first 6 elements is 252. Find the first element in this sequence. I actually…
Tarek Zoabi
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What is the value of $\sum_{k=1}^{60} A_k$ knowing that $A_{n+1}+(-1)^nA_n = 2n-1$ for all $n$?

This sequence meet $A_{n+1}+(-1)^nA_n = 2n-1$ condition. What is the result of the sum from $A_1$ to $A_{60}$?
Shubham Kumar
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Find the value of $m$ given that the sum of the first $m$ terms is equal to the sum of the first $(m+1)$ terms.

The first term of an arithmetic progression is $100$ and the common difference is $-5$. The answer should be $20$, but how? Please explain the solution.
Saud Kamran
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Need to find initial term and the common difference A.P

What's given: Sum of the first 6 terms is equal to -12 and the sum of the LAST 8 terms of progression is equal to -224 I'm required to find the initial term and the common difference
guy
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