# All questions with tag [math: arithmetic-progressions]

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## $\sum \cos$ when angles are in arithmetic progression

Possible Duplicate: ยข How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? Prove $$\cos(\alpha) + \cos(\alpha + \beta) + \cos(\alpha + 2\beta) + \dots + \cos[\alpha + (n-1)\beta] = \frac{\cos(\alpha + \frac{n-1}{2}\beta) \cdot \sin\frac{n\beta}{2}}{\sin\frac{\beta}{2}}$$
2022-07-11 23:40:04
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## Problem about sum of arithemtic progression and geometric progression

Question: A math series has an usual distinction of $1$ and also a geometric series has an usual proportion of $3$. A new series is created by including equivalent regards to these 2 developments. It is considered that the 2nd and also 4th regard to the series are $12$ and also $86$ specifically. Locate, in regards to $n$, the $n^{th}$ term,...
2022-06-12 11:14:21
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## Why is sum of a sequence $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?

Is there a means to confirm that the amount of the math development $a_1, a_2, \dots, a_n$ can be computed by $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?
2022-06-08 05:50:57
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## How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

Just how can we summarize $\sin$ and also $\cos$ collection when the angles remain in math development? As an example below is the amount of $\cos$ collection : $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$ There is a mild distinction in ...
2022-06-03 18:17:32
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## Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Evidently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. Just how? What's the evidence? Or possibly it is self noticeable simply considering the above? PS : This trouble is called "The amount of the first $n$ favorable integers".
2019-05-09 11:19:32