Why do $\sum_{i=0}^3 (-2^i)$ and $\sum_{i=0}^3 ((-2)^i)$ give different values?

Question

On my TI-84, I have noticed something weird. I have: $$\sum_{i=0}^3 (-2^i) = -15$$ and $$\sum_{i=0}^3 ((-2)^i) = -5$$

Does anybody know why these sum to a different number?

Correct me if I am wrong but isn't $-2^i$ the same as $(-2)^i$?

Answer 1

In standard order of operations, the unary minus happens after exponentiation. So $-2^k=-(2^k),$ not $(-2)^k,$ which are different when $k$ is even.

In the standard order of operations, exponentiation precedes all the standard operations, so, also, in general:

$$a+b^n=a+(b^n)\neq (a+b)^n\\ a\cdot b^n=a\cdot(b^n)\neq (a\cdot b)^n$$

---

It’s worth noting that the standard order of operations is just a human convention, letting us shorten expression. Abstractly, we should write all the parentheses. So, instead of $1+2\cdot 3+4,$ we’d write:

$$(1+(2\cdot 3))+4$$

That would suck though, so we find a better way to communicate consistently and meaningfully.

These rules are not mathematical laws - they aren’t proven, they are just how we communicate.

There are things we can (and should) prove about the rules. We want expressions under our rules to be interpreted unambiguously, for example.

But there are other language rules we could use that communicate equally unambiguously. And which set of rules we use is a human choice, like choosing between English and German.

Answer 2

You can understand these notations ("meat" of your question/problem) as follows:

• Interpretation of $-2^n$

$$-2^n=(-1)\times 2^n=-\left(2^n\right)$$

• Interpretation of $\left(-2\right)^n$

$$\left(-2\right)^n=\left((-1)\times 2\right)^n=(-1)^n \times 2^n$$

Now, I think you can observe the difference comfortably:

$$\begin{align}(-1)\times 2^n&≠(-1)^n\times 2^n \\ \iff (-1)&≠(-1)^n\end{align}$$

Then, you can clearly see that, if $n$ is an even natural number, then the equality doesn't hold. The equality holds if and only if when $n$ is an odd natural number.

Answer 3

No, the first expression means summation of $-(2^i)$. I think the meaning of the second is quite clear.

Answer 4

In the first example your calculator finds

$$-(1+2+4+8)=-15$$

whilst in the second, because of the brackets it finds $$1-2+4-8=-5$$

Answer 5

Consider the power of something as $\square^i$. Power belongs to the things in the square.

• The first sum: $-\boxed{2}^{~i}$
• The second sum: $\boxed{-2}^{~i}$

Can you differ them now?

Answer 6

$\sum_{i=0}^3 (-2^i) = -2^0-2^1-2^2-2^3=-1-2-4-8=-15$

$\sum_{i=0}^3 ((-2)^i) = (-2)^0+(-2)^1+(-2)^2+(-2)^3=1-2+4-8=-5$