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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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Calculate approximate cost of service based on companies' revenue

We create bespoke web designs and we always get asked how much the designs will cost without even knowing the job. Some people don't like to be asked for their budget, some can't understand how hourly rate works either, so, to simplify things we…
Thomas
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Compute the value of the sum $7\cdot 11+11\cdot 15+15\cdot 19+\cdots+95\cdot 99$

Solution: My attempt: =$9^2-2^2 +13^2-2^2+17^2-2^2...97^2-2^2$ =$(9^2+13^2+17^2...97^2)-2^2( 23 )$ Focus on the first part. 81+169+289+441...Here first-order differences are 88, 120, 152... and the second-order difference is common to be 32.…
Sherlock Watson
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Sum of the number of terms in an arithmetic sequence formula question.

The nth term formula is that $a_n = a+(n-1)d$ How does this formula convert to the formula for the number of terms in an arithmetic sequence which is equal to $$\dfrac{\text{last term-first term}}{ \text{common difference}}+1$$ In other words, how…
user57928
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Proof of the arithmetic sequences general formula (not using the 1st term) to find the n-th

Can anyone explain on how come the second form of the formula is valid? Is there anyway to prove its validity without plugging in the numbers? Formula $1$ : $a_n=a_1+(n-1)d$ Formula $2$ : $a_n=a_m+(n-m)d$
D.Mantas
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Sequences and series(Arithmetic and Geometric progression)

Anyone can help me solve this question？ The first three of four integers are in an a.p. and the last three are in g.p. Find these four numbers, given that the sum of the first and the last integers is 37 and the sum of the two integers in the middle…
Lynn
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if the sum of m terms of an A.P to n terms is $m^2$ to$n^2$ then show that the $m$th term to $n$th term is $2m-1$ to $2n-1$

QUESTION: if the sum of m terms of an A.P to n terms is $m^2$ to $n^2$ then show that the $m^{th}$ term to $n^{th}$ term is $2m-1$ to $2n-1$ MY ATTEMPT: using the formula for sum of n terms of an A.P where $a$ is the first term , $d$ is the…
Shash
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arithmetic sequence (Algebra 1, Khan Academy)

I am learning Mathematics from Khan Academy. I see this explicit formula in arithmetic sequence: f(n) = 3 - 4(n-1) I have to find which term in the sequence = -65 I found it out by crude methods of calculating f(10), f(20) etc, it is 18th term. My…
arnuld
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Problem related to sum of an Arithmetic Progression.

If there are $(2n+1)$ terms in A.P.,prove that the ratio of the sum of odd terms and the sum of even terms are in the ratio of $(n+1):n.$ The main confusion I am facing is that n is not specified here. I feel that It could be anything from a…
Arunabh
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Arithmetic Progression proof for sum of an integer

Question:- The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that resulting sum is the square of an integer.  MyApproach: Let d be the…
user517784
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How do I proceed with this?

$$\sum_{n=1}^N\prod_{k=1}^K\sum_{m=1}^M\left[\frac{bk^3n^5-cm^2}{n^2+ak^4}\right]$$ I must make it work in a C programme , for my C class , and we will give number for the $a, b ,c , N , M, K$ in the programme. But how will I write it? First the…
Constantina C.
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Defining a rigorous equation for the distance

So this question came up in my exam and luckily I had time so I used trial and error to solve it however I am looking for a rigorous equation to solve it. The question statement is that there is an odd number of stones along a road and each of them…
Prakhar Nagpal
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Arithmetic progression determination

I love math but I really suck at and I definitely don't know cool tricks (I stem more from the science part of STEM) so I may even have the wrong term in the title. That said here is my problem. I'm writing a function for a database and have an…
Lux Claridge
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Does not depend on n

Suppose that $S(k)$ is the sum of the first k terms of an arithmetic sequence with common difference 3. If the value of$$S(3n) / S(n)$$ does not depend on n, what is the 100th term of the sequence? What does "does not depend on n" mean and how do I…
SuperMage1
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Common term between arithmetic progression

How do I mathematically show that the common terms between the series $3+7+11+....$ and $1+6+11+....$ form an arithmetic progression without actually finding all the individual terms. How does the LCM of the common differences of the given series…
Sooraj S
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Using intuition when proving things

Let's say we have an arithmetic progression: $a_{n+1} = a_n + k$. Fair enough, now I want to prove $a_{n+m} = a_n + mk$. This is how I do it: $$a_{n+2} = a_{(n+1)+1} = a_{n+1} + k = a_n + k + k = a_n + 2k$$ $$a_{n+2} = a_n + 2k$$ Oh well, now I…
user3600124
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