Prove by induction the sum of complex numbers is complex number

[cbarker1]

Homework Statement

Prove by induction that for $z$ is a complex then $\sum_{i=1}^{n}z_i$

Relevant Equations

$P(0)$ is true.
If $P(k)$ is true, then $P(k+1)$ is true

See the work below:

I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.

 

[topsquark]

Homework Statement:: Prove by induction that for #z# is a complex then #\sum_{i=1}^{n}z_i#
Relevant Equations:: P(0) is true.
If P(k) is true, then P(k+1) is true

See the work below:

I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.

Hint: Enclose your code in ##, not #.

The steps in your induction are correct. I must ask, though: You know that if y = 0 that z is still a complex number, right?

-Dan

 

[cbarker1]

Hint: Enclose your code in ##, not #.

The steps in your induction are correct. I must ask, though: You know that if y = 0 that z is still a complex number, right?

-Dan

yes. I do.

 

[FactChecker]

The steps are all there, but your notation is bad. You are using $z$ for too many different things. You should distinguish between the sum of n versus the sum of n+1. Call the sum of n $s_n$ and the sum of n+1 $s_{n+1}$.
Also, it would be a little more clear if, before the last line, you said that $s_n + z_{n+1}$ is the sum of two complex numbers so it is complex.

 

[PeroK]

In general:

If $$a, b \in S \implies a + b \in S$$ then:$$a_1, \dots a_n \in S \implies \sum_{i=1}^{n}a_i \in S$$