How do I solve the following recurrence relation?

$$ T(n) = 1 + \sum_{j=0}^{n-1} T(j). $$

I thought of solving it by generating its recursion tree. I found that the height of the tree would be equal to $n$ but wasn't able to find the cost of each level.


1 Answer 1


Here are several ways to solve your recurrence relation.


Anyone with enough experience in computer science might recognize your recurrence as the one satisfied by $T(n) = 2^n$. Given this guess, you can verify it by summing the appropriate geometric series: if $T(m) = 2^m$ for $m < n$ then

$$ T(n) = 1 + \sum_{m=0}^{n-1} T(m) = 1 + \sum_{m=0}^{n-1} 2^m = 1 + (2^n-1) = 2^n. $$

Computing the first few values

Another approach which is very useful is to compute a few values of the sequence: $$ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... $$ If we're lucky, as in this case, we might spot a pattern. If not, it's always a good idea to consult the on-line encyclopedia of integer sequences. A quick search would reveal the sequence A000079.

Creative telescoping

In the method of creative telescoping, we try to find a combination of terms that results in major cancellation. In our case, we can try $T(n+1)-T(n)$:

$$ T(n+1)-T(n) = \sum_{m=0}^n T(m) - \sum_{m=0}^{n-1} T(m) = T(n), $$ and so $T(n+1) = 2T(n)$. Unrolling this rule, we deduce $T(n) = 2^n T(0) = 2^n$.


Another way to look at the preceding trick is to try to "back-substitute" the recurrence inside itself:

$$ T(n) = 1 + \sum_{m=0}^{n-1} T(m) = 1 + \sum_{m=0}^{n-2} T(m) + T(n-1) = 2T(n-1), $$

from which we can proceed as before.

Generating series

A very general method is that of generating series. Let $G(x) = \sum_{n=0}^\infty T(n) x^n$. Since $1/(1-x) = \sum_{n=0}^\infty x^i$, multiplying by $1/(1-x)$ is like computing "running sums": $$ \frac{G(x)}{1-x} = \sum_{n=0}^\infty \sum_{m=0}^n T(m) x^n = \sum_{n=0}^\infty (T(n+1)-1) x^n = \\ \frac{G(x)-T(0)}{x} - \sum_{n=0}^\infty x^n = \frac{G(x)-1}{x} - \frac{1}{1-x}. $$ From this it is not hard to deduce $$ G(x) = \frac{-1/x-1/(1-x)}{1/(1-x)-1/x} = \frac{-(1-x)-x}{x-(1-x)} = \frac{1}{1-2x} = \sum_{n=0}^\infty 2^nx^n, $$ and so $T(n) = 2^n$.

Recursion tree

It is also possible to deduce the value of $T(n)$ using a recursion tree. Each path in the recursion tree for $T(n)$ corresponds to a sequence of decreasing numbers $n_0 = n, \ldots, n_\ell$, where $n_\ell \geq 0$ (possibly $\ell = 0$); this is because all leaves are labeled by $1$. The set of possible decreasing sequences is in one-to-one correspondence with subsets of $\{n-1,\ldots,0\}$, because once we know which numbers appear, we also know their order. The number of subsets of $\{n-1,\ldots,0\}$ is exactly $2^n$.

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    $\begingroup$ I really appreciate your experience. $\endgroup$ Commented Mar 28, 2018 at 15:40

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