Sum of Sequence of Products of Consecutive Integers/Proof 1
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Theorem
\(\ds \sum_{j \mathop = 1}^n j \paren {j + 1}\) | \(=\) | \(\ds 1 \times 2 + 2 \times 3 + \dotsb + n \paren {n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} \paren {n + 2} } 3\) |
Proof
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
- $\ds \forall n \ge 1: \sum_{j \mathop = 1}^n j \paren {j + 1} = \frac {n \paren {n + 1} \paren {n + 2} } 3$
Basis for the Induction
$\map P 1$ is true, as this just says $1 \times 2 = 2 = \dfrac {1 \times 2 \times 3} 3$.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $\ds \sum_{j \mathop = 1}^k j \paren {j + 1} = \frac {k \paren {k + 1} \paren {k + 2} } 3$
Then we need to show:
- $\ds \sum_{j \mathop = 1}^{k + 1} j \paren {j + 1} = \frac {\paren {k + 1} \paren {k + 2} \paren {k + 3} } 3$
Induction Step
This is our induction step:
\(\ds \sum_{j \mathop = 1}^{k + 1} j \paren {j + 1}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^k j \paren {j + 1} + \paren {k + 1} \paren {k + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k \paren {k + 1} \paren {k + 2} } 3 + \paren {k + 1} \paren {k + 2}\) | Induction hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k \paren {k + 1} \paren {k + 2} + 3 \paren {k + 1} \paren {k + 2} } 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {k + 1} \paren {k + 2} \paren {k + 3} } 3\) |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \ge 1: \sum_{j \mathop = 1}^n j \paren {j + 1} = \frac {n \paren {n + 1} \paren {n + 2} } 3$
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Method of Induction: Exercises $\text {II}$: $2$
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $1 \ \text {(c)}$