Represent $1^5+2^4+3^3+4^2+5^1$ in sum notation

Question

I imagine the sum notation for

$$1^5+2^4+3^3+4^2+5^1$$

Would look something like

$$ \sum x^y,\ x=1 \text{ to } 5,\ y=5 \text{ to } 1 $$

Is this correct or am I missing something?

Answer 1

Observe that in each term, the sum of the base and the exponent is $6$. Thus, if the base is $k$, the exponent is $6 - k$. Since the base increases from $1$ to $5$, we obtain $$\sum_{k = 1}^{5} k^{6 - k}$$

Answer 2

Two ways I can think of:

$$\sum_{k=1}^5 k^{6-k}$$

$$\sum_{a,b\in\mathbb{N}_{\ge1}: a+b=6}a^b$$

Answer 3

$\sum_{i=1}^5 i^{6-i}$ is equal to your sum

Answer 4

One way in which it would often be done is $$ \sum_{x=1}^5 x^{(1+5)-x}. $$ If one were to write $$ \sum_{(x,y)=(1,5)}^{(5,1)} x^y $$ perhaps that would be understood, but I might write something after it like this:

where $(x,y)$ runs through the set of integers points on a straight line connecting the two extremes

or the like. With a suitable comment like that I would expect it to be understood.