1
$\begingroup$

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit?

By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.

$\endgroup$
  • $\begingroup$ Try to determine the $n$th term in the sequence in terms of $n$, $a_1$ and $a_2$. You should notice that $a_n=f(n)a_1+g(n)a_2$, where $f,g$ are some functions related to $2^n$ (and you can prove this by induction). Try to prove that $f(n)\to\frac{1}{3}$ and $g(n)\to\frac{2}{3}$. $\endgroup$ – Peter Huxford Sep 12 '14 at 7:35
  • 3
    $\begingroup$ Please use the word sequence instead of series. They have different meanings in maths. $\endgroup$ – user37238 Sep 12 '14 at 8:17
0
$\begingroup$

We have $$2a_n-a_{n-1}-a_{n-2}=0$$

Using Recurrence Relation formula , $$a_n=A+B\left(-\frac12\right)^n$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.