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How do I find the sum of a sequence whose common difference is in Arithmetic Progression ?

Like in the following series :-

$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91$

And also how to find it's $n^{th}$ term ??

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  • $\begingroup$ This is the triangular numbers. You just get a quadratic term on the eventual summation. This is a good exercise in Algebra, I think you should try this, starting from the normal formula for arithmetic progression and the derivation of that formula. $\endgroup$ – астон вілла олоф мэллбэрг Oct 10 '16 at 9:09
  • $\begingroup$ if the difference is not the same at every step you can't say it's a common difference. $\endgroup$ – mercio Oct 10 '16 at 9:36
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You might be interested to know that if the differences of consecutive terms of a sequence are given by some polynomial, i.e $a_{n+1}-a_n=P(n)$, for some polynomial $P$, then the terms of the sequence themselves are given by a polynomial, with one greater degree. As in your case, the differences of terms are in A.P, i.e the differences are given by a linear polynomial: $$a_{n+1}-a_n=P(n)=n+1$$

So, the general term of the sequence will be given by a quadratic polynomial $an^2+bn+c$. All you have to do is find the constants $a,b,c$ which you can do using the first three terms of the sequence. As for summation of this sequence, use the known summations $\sum_{k=1}^nk$ and $\sum_{k=1}^nk^2$.

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The sequence that you are talking about is a quadratic sequence. A quadratic sequence is a sequence of numbers in which the second difference between any two consecutive terms is constant (definition taken from here).

The difference of consecutive terms in your sequence forms an arithmetic progression $2,3,4,5,\dots$ with common difference of $1$. Since the sequence is a quadratic sequence, the $n^{th}$ term of the sequence is given by a quadratic polynomial: $$T_n = an^2 + bn + c$$ As stated by Siddhant in his answer, you could just plug in $n$ as $1,2$ and $3$ to get three equations in three variables and get the values of $a,b$ and $c$. However, we could use a rather generalized formula for getting the equation for $n^{th}$ term of the sequence (you should try deriving this formula): $$T_n = \left(\frac{d_0}{2}\right)n^2 + \left(d - \frac{3\cdot d_0}{2}\right)n + (a + d_0 - d)$$ where $a$ is the first term of the sequence, $d = T_2 - T_1$ i.e. difference between first two terms of the sequence and $d_0$ is the second difference between any two consecutive terms of the sequence. In your case, $a = 1$, $d = 2$ and $d_0 = 1$. Plugging in these values in the equation yields $$T_n = \frac{1}{2}n^2 + \frac{1}{2}n$$

For finding the sum: $$\sum_{i = 1}^{n}T_i$$ $$=\sum_{i = 1}^{n}\left(\frac{1}{2}i^2 + \frac{1}{2}i\right)$$ You can solve this using the already known summations $\sum_{i = 1}^ni^2$ and $\sum_{i = 1}^ni$.

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In such a case, every generic term of the sequence can be represented as

$An^2 + Bn + C$

where A B C are fixed values and n is the position of the term. If the difference of difference of the terms were in an ap, the generic term would be

$An^3 + Bn^2 + Cn + D$

Where A B C and D are fixed values.

This will keep on going till the differences are in AP.

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$$ \begin{align} & \text{Let } a = a_{0} \space\,,\space d = \frac{1}{n} (a_{n} - a_{n-1}) \space\Rightarrow\space a_{n} = a_{n-1} + d \cdot n \quad \colon n \ge 1 \\ & \small \color{blue}{ (a=0 \,, d=1) \Rightarrow a_{n} = a_{n-1} + n = \{ \text{1, 3, 6, 10, ...} \} } \\ & \\ & a_{n} = d (n) + a_{n-1} = d (n) + d (n-1) + a_{n-2} = d (n) + d (n-1) + \text{...} + d (2) + d (1) + a \Rightarrow \\ & a_{n} = a + d \sum_{k=1}^{n} (k) = a + \frac{d}{2} n (n+1) \\ & \small \color{blue}{ (a=0 \,, d=1) \Rightarrow a_{n} = \frac{1}{2} n(n+1) = \{ \text{1, 3, 6, 10, ...} \} } \\ & \\ & \sum_{k=1}^{n} a_{k} = \sum_{k=1}^{n} \left( a + \frac{d}{2} k(k+1) \right) = a \cdot n + \frac{d}{2} \left[ \sum_{k=1}^{n} k + \sum_{k=1}^{n} k^2 \right] \Rightarrow \\ & \sum_{k=1}^{n} a_{k} = a n + \frac{d}{2} \left[ \frac{n(n+1)}{2} + \frac{n(n+1)(2n+1)}{6} \right] = a n + \frac{d}{6} n(n+1)(n+2) \\ & \small \color{blue}{ (a=0 \,, d=1) \Rightarrow \sum_{k=1}^{n} a_{k} = \frac{1}{6} n(n+1)(n+2) = \{ \text{1, 4, 10, 20, ...} \} } \end{align} $$

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Empirically:

By educated guess, as the differences grows as $n$, the terms grow as $n^2$. Taking the ratios $T_n/n^2$, you see them converging to $1/2$.

Then the sequence of $2T_n-n^2$ is

$$1,2,3,4,5,6,7,\cdots$$

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