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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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Arithmetic Progression, Tn and Tm proving sum

How can I prove this? If in an A.P., Tm=n, Tn=m, prove d=-1.
Chandan Das
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From Arithmetic Progression, Sums of n terms using formulae: $T(n)=a+(n-1)d, S(n)=n/2[2a+(n-1)d]$

How to find the below questions answer? I have tried but can't find anything? I think it's too much tough! Please anybody can help me? Find the sums indicated below: 3+6+9+...+300
Chandan Das
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Arithmetic sequence question

The question is: the first three terms of an arithmetic series $c_{n}$ are $$a(1+b), a(1+3b),a(1+5b)$$ I needed to find the common difference in terms of $a$ and $b$ and then find the expression for $c_{n}$. The final part I struggled with where I…
Nikhil Patel
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Finding $c$ given $a, ab, abc$ are in A.P

If a, b, c are distinct nonzero integers such that a, ab, abc are in A.P then find the value of c. The answer is one among this : 1, 2, 3 or 4. I tried $$\frac{a+abc}{2}=ab$$ But solving doesn't seem to give a numerical answer independent of b.
H G Sur
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How do I solve this arithmetic sequences problem?

Let ${a_n}$ and ${b_n}$ be two different arithmetic series such that $(3n+1)a_n=(2n-1)b_n$ for all positive integers $n$. Let $A_n=a_1+a_2+...+a_n$ and $B_n=b_1+b_2+...+b_n$. what is the value of $\frac{A_9}{B_6}$? How do I do this? There isn't any…
user406996
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Find the 20th member of an arithmetic progression

The progression is increasing arithmetic I am given that $a_2+a_3+a_4+a_5=34$ and that $a_2\cdot a_5=52$ I find that $a_5=13$ or $4$ it can't be $4$ so when i try with $a_5=13$ the $a_{20}=43$ and the answer is $58$
yolo expectz
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Find the number of terms in this AP

The AP is defined as follows: $$ 7 + 9 + ... + (2n + 1) $$ If I remember correctly the answer given was $n-2$. But I thought that since $n$ was defined as the number of terms in the progression, the answer should always be $n$. How can the answer be…
Sam Houston
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Finding common terms of two or more arithmetic sequences

Suppose I have three sequences $\{1,4,7,....,2998\}$ , $\{1,3,5,7,9,11,....,3001\}$ and $\{1,6,11,....,4001\}$ . How can i find the number of common terms among them ? For Example , If I have 1,3,5,7,9,11,13,15,17 and 19 1,4,7,10,13,16 and…
psil123
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If the sum of the first $n$ terms of an A.P is given $Sn$= $2n^2+5n$. Find the $n$th term of the A.P?

If the sum of the first $n$ terms of an A.P is given $Sn$= $2n^2+5n$. Find the $n$th term of the A.P ?
changer
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Find whether ribbon is sufficient

Joe cuts pieces of ribbon for a decoration according to a pattern, such that the first piece is 20 cm long, the second piece is 25 cm long and the third piece is 30 cm long. The longest piece of ribbon she requires for the decoration is of length…
user360471
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Find the value of $\sum_{n=1}^k u_n$, arithmetic series

An arithmetic series has first term $80$ and common difference $5$. The sum of the first $25$ terms of the series is $3500$. The $n$th term of the series is $u_n$. Given that $$33\left(\sum_{n=1}^{25} u_n-\sum_{n=1}^k u_n\right)=67\sum_{n=1}^k…
Thomas Winkworth
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Find terms number of arithmetic progression.

I had an exam today, within the exam, this question was the hardest. If we have a arithmetic progression, its number of terms is $even$, total of it's $even$ terms = $30$, total of it's $odd$ terms = $24$. the difference between the last term and…
Slavi
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I wanna figure out the numbers in an arithmetic sequence question

The sum of three consecutive terms in an arithmetic sequence is 21 and the product of the two extreme numbers is 45. Find the numbers. How would we go on about finding this?!
SomeDev
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Why is the sum of the next five sums greater than the sum of the first five in the following Arithmetic Progression?

The question: In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is -112. I was able to solve the actual problem easily. But the strange thing is that the common difference…
Aayush Agrawal
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arithmetic progression, problem.

A company will distribute $\$46,000$ in bonuses to their top ten vendors. The last winner from the list will receive $\$1000$ and the difference in money between sellers on classifieds should be constant. Find the bonus for each worker. You know how…
Splitzer
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