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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, finding the nth term.

The sum of the 1st n terms, of an AP is $S_n=n^2-3n$. Write down the 4th term and find an expression for the $n$th term. Will the 4th term be $t_4= a+3d$?
Tesha Caesar
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Arithmetic progression. find in terms of n

Find in terms of n: $$\sum_{r=1}^{n} 2r-1$$ and $$ \sum_{r=0}^{n} 3r+3 $$ I tried using summation but they said in terms of n and then I expanded substituting from numbers 1 and 4.
Tesha Caesar
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Arithmetic Progression first term from Sum and Common Difference

The sum of the first 50 elements of a set is 6925 with a common difference of 5. What is the first element of the set I know how I would usually find the first term of an AP, but I cannot work out how to work out what the 50th term is from the sum…
Dhatsah
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Arithmetic Progression with dynamic common difference

While preparing for my recent exams I noticed some interesting AP problems which I was not able understand. The problem was that the common difference was itself in arithmetic progression For Example: 1,3,6,10,15... You can thus clearly see that…
Palash Taneja
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How mamy trailing zeroes in 1000! Using G.P. or A.P.?

I figured out the way to calculate the number of trailing zeroes in 1000! Where ! Denotes factorial but I dont know how to use A.P. OR G.P. Original question:1000! is divisible by 10^n , find the largest possible integer value of n. My way to…
Kartik Watwani
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A tricky arithmetic progression question

In a question I have to prove that if $\log_l x, \log_m x, \log_n x$ are in AP where $x \neq 1$ and $x > 0$, prove that $$n^2=(l \cdot n)^{\log_l m}$$ My tries: I first converted every term to natural logarithm so I got ln (x)/ln (l), ln…
Kartik Watwani
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A tricky arithmetic progression problem

In a question I have to prove that if log (base l of x), log (base m of x), log (base n of x) are in AP where x doesnt equals 1 and x is positive, prove that n^2=(l*n)^(log base l of m)> My tries: I first converted every term to natural…
Kartik Watwani
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Demonstrating $u_0$ and common difference before calculating them in arithmetic progression

I'm getting stuck with a type of exercise on arithmetic progressions I never done before. $\{u_n\}$ is an arithmetic progression: $u_1+ u_2 + u_3 = 9$ $u_{10}+ u_{11} = 40$ I have then to prove that $u_0$ and the common difference $r$ are…
Dhazard
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How to calculate an arithmetic pattern which follows squaring?

Recently, I've come accross a situation where I needed to calculate a sum of the following pattern: $$x^2+(x+1)^2+(x+2)^2+...+(x+n)^2$$ How do I calculate the sum of it?
Dipanjan Das
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Largest possible common difference of an arithmetic progression given its three terms (not necessarily adjacent)

How can I find the largest possible common difference of an arithmetic progression given that its three terms (not necessarily adjacent) are $0.37$, $9$ and $\frac{71}{7}$?
Overdrowsed
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Prove the following statment $\frac{S_m-S_n}{S_{m+n}}=\frac{m-n}{m+n}$ if $S_m$, $S_n$ and $S_{m+n}$ are arthmetic series

I proved this statment using formulas for n-th term of aritmetic sequence and for arithmetic series. Does anyone know any other proof?
Theo Gray
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(Sum of multiples of $3$ between $1$ and $100$) $-$ (Sum of multiples of $3$ between $5$ and $95$)

$m$ is the sum of all multiples of $3$ between $1$ and $100$. $n$ is the sum of all multiples of $3$ between $5$ and $95$. what is $m-n$?
Maybeline Lee
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Prove that $ u_1 + u_5 = 2u_3 $.

$u_n$ is an arithmetic progression which is an increasing sequence. Prove that $$ u_1 + u_5 = 2u_3 $$ is it correct to rely on the law that states that $a + c = 2b$.
AnisLaib
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Simple Arithmatic progrssion problem

A man saves Rs $32$ during first year ,$36$ in the next year $40$ in $3$rd year .if he continue his savings in this sequence,in how many years he saves $2000$ Rs. Rs=currency
Epic5tv
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What is a2-a1 : b3-b1 in this AP?

X a1 a2 Y X b1 b2 b3 Y what is a2-a1 : b3-b1 in this AP? i was getting an answer of 2/3. don't know if it's correct.
user278077
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