The image you uploaded contains a series of mathematical expressions involving summations and functions $f(n)$, $f(1)$, $f(2)$, and so on. Let me transcribe the contents to analyze and define the function $f(n)$.

Transcription of the Image

1. The image contains the following expression:

$\sum_{k=1}^{n} \sum_{i=1}^{n} f(i+k-1) - \sum_{k=1}^{n} \sum_{i=1}^{n} f(i) = 0$

2. The equation can be expanded to:

$\sum_{k=1}^{n} \left( \sum_{i=1}^{n} f(i+k-1) \right) = \sum_{k=1}^{n} \left( \sum_{i=1}^{n} f(i) \right)$

Simplification and Analysis

• The left side of the equation involves shifting the index $i$ by $k-1$ inside the function $f$. This creates a sequence of terms where the argument of $f$ is incremented by $k-1$ for each $k$.

• The right side sums the same function $f(i)$ over $i$ from 1 to $n$, but without shifting.

For this equality to hold for any arbitrary $n$, it suggests that $f$ is likely a function that does not change with respect to a shift in its argument. One natural conclusion is that $f(i)$ is a constant function, i.e., $f(i) = c$ for some constant $c$.

Conclusion

Given the equation, the most reasonable definition of $f(n)$ is that it is a constant function:

$f(n) = c$

where $c$ is a constant.

This satisfies the condition because the sum of constant values on both sides of the equation would be equal.

Questions and Tip

1. What assumptions are we making about the function $f(n)$ when concluding it's a constant?
2. How would the analysis change if $f(n)$ were a linear function instead of a constant?
3. Could $f(n)$ be periodic, and if so, how would that impact the equation?
4. How can this form of summation be applied in more complex scenarios, such as varying the bounds of the summation?
5. What would be the implications if $f(n)$ were a non-continuous function?

Tip: When dealing with summations that equate, considering the nature of the function (whether it's constant, linear, or periodic) is crucial for simplifying and understanding the equation.