Consider the sequence $(T_n) = 1, 4, 9, 16 ...$
In and exercise I'm trying to solve I'm asked to:
A) Show that $(T_n)$ is defined by $T_n=1+3+5+7+...+2n-1$
B) Prove that the general term for $T_n$ is $n^2$
Assuming the A) is true, I can set $1+3+5+7+...+2n-1=n^2$ , and then prove by induction that $(T_n)$ is indeed of general term $n^2$.
But how can I prove A)?
Edit: What other ways are there to prove this?
A nice way to determine the sum : $$S=1+3+5+\cdots (2n-1)$$
Write the sum in reverse $$S=(2n-1)+\cdots +5+3+1$$
Now, you see by adding the columns that $$2S=2n\cdot n$$ holds, which immediately gives $S=n^2$