I have a problem with this type of recurrence equation.

Find the solution of recurrence equation:


$$T(n+1)=T(n)+2n , \quad \forall n\geq 1$$


I tired to Solving Recurrences Using Substitution









I Sum up all these, and I got that:


but i'm stack there

  1. Any help would be highly appreciated.
  2. Can we Solving Recurrences Using Other methods suck like :

     The Characteristic Equation

There are many methods. We describe one a little different than the one you were carrying out.

The homogeneous recurrence $T_{n+1}=T_n$ has as only solution $a_n=A$. This is immediate. But to answer your last question, it could be thought of as using the characteristic equation method. The characteristic equation is the very uninteresting $x=1$. So the general solution of the homogenous recurrence is $A(1)^n$.

Now that we have the general solution of the homogeneous recurrence, we look for a particular solution of the original recurrence. Guess that it will be $T_n=an^2+bn+c$. Substituting in the original recurrence, we get $$a(n+1)^2+b(n+1)+c=an^2+bn+c+2n.$$ Expand, and compare coefficients. We get $a=1$ and $a+b=0$, giving particular solution $n^2-n$.

Thus the general solution of our recurrence is $T_n=A+n^2-n$. To evaluate $A$, use the initial condition $T_1=2$.

Remark: You were very close to finding a particular solution. Let us take the expression you got at the end, take out the common factor $2$, and write it down backwards. We get $$2(1+2+3+\cdots+(n-1)).$$ Recall that $1+2+3+\cdots +m=\frac{m(m+1)}{2}$. Putting $m=n-1$, we get $2\frac{(n-1)(n)}{2}$, that is, $n(n-1)$. That is exactly the particular solution obtained in the answer above.

  • $\begingroup$ Thanks sir, but what is the exact answer is T(n)=n(n−1) or T(n)=n(n−1)+2. i'm confusing could you please explain more $\endgroup$ – Adam Apr 30 '14 at 18:20
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    $\begingroup$ You need to evaluate $A$, using $T_1=2$. Substituting, we find that $A=2$, so the solution for the given initial condition is $2+n(n-1)$. $\endgroup$ – André Nicolas Apr 30 '14 at 20:29

You should merely recall that


Then, I think, $T(n)=n(n-1)+2$.


As a general rule if $T(n+1) - T(n)$ is a polynomial of degree $k$ then $T(n)$ is a polynomial of degree $k+1$. In this case, $T(n+1)-T(n)=2n$, a 1st-degree polynomial, so $T(n)$ is a 2nd-degree polynomial.

Thus, set $T(n)=A n^2 + B n + C$ where $A, B, C$ are coefficients to be determined.

Now plug that quadratic expression above into your recurrence relation, and you will end up with a system of equations in A, B, and C. Solve that system and you are done.


Related problems: (I), (II). Here is an approach

$$ T(n+1)=T(n)+2n , \quad \forall n\geq 1 \implies \sum_{k=1}^{n}(T(k+1)-T(k))=\sum_{k=1}^{n}2k $$

$$ \implies T(n+1)-T(1)=n(n+1) \implies T(n+1)=n(n+1)+2 $$

$$ \implies T(n)=(n-1)n+2=n^2-2n+2. $$

If you are interested in other approaches see here.

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    $\begingroup$ you using Telescoping series Great idea $\endgroup$ – Adam Apr 30 '14 at 18:36
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    $\begingroup$ @Adam: Yes, and in this case it is very helpful. $\endgroup$ – Mhenni Benghorbal Apr 30 '14 at 18:36

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