What is the conjecture being proved in the video example?

Conditions

• n is a natural number.
• The left side is the sum of consecutive even integers starting from 2.

Reasoning, step by step

1. Identify the sequence of even integers: 2, 4, 6, ..., 2n.
2. Observe the pattern for small n (e.g., $n=1$: $2=1*2$, $n=2$: $6=2*3$).
3. Formulate the general conjecture: Sum = $n(n+1)$.
4. Define the proposition $P(n)$ as this equality.

Example

The video displays the formula '$2 + 4 + 6$ + ... + $2n = n(n+1)$' prominently at the top of the screen throughout the induction proof. The narrator introduces it as the conjecture to be proved.

Common misconceptions

• Confusing the sum of odd integers with even integers.
• Thinking the formula is $n^2$ or $n(n-1)$.

Watch the explanation

Explore next

• How is this conjecture proved by induction?
• What is the base case for this conjecture?