Question

Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3

Accepted Answer

Step 1: Define the statement to prove by induction. Let $$ P(n)  be $$the statement $$ 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + n(n + 1) = \frac{n(n + 1)(n + 2)}{3} . We $$want to prove $$ P(n)  is $$true for all positive integers $$ n . $$Step 2: Verify the base case $$ n = 1 . $$Substitute $$ n = 1 $$ into both sides of the equation and check if the left-hand side equals the right-hand side. Step 3: Assume the induction hypothesis $$ P(k)  is $$true for some positive integer $$ k $$, that is, assume $$ 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) = \frac{k(k + 1)(k + 2)}{3} . $$Step 4: Prove the statement $$ P(k + 1)  is $$true using the induction hypothesis. Start with the left-hand side of $$ P(k + 1) $$: $$ 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) + (k + 1)(k + 2) . $$Substitute the induction hypothesis for the sum up to $$ k(k + 1) . $$Step 5: Simplify the expression obtained in Step 4 and show that it equals the right-hand side of $$ P(k + 1) $$, which is $$ \frac{(k + 1)(k + 2)(k + 3)}{3} . $$This completes the induction step and proves the statement for all positive integers $$ n .$$

Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3

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Key Concepts

Mathematical Induction

Summation of Sequences

Algebraic Manipulation