Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false:

4 x 6 +5 x 7 + 6 x 8+...+4n(4n+2)= 4(4n+1)(8n+7) / 6

Answer 1

`\text{Proof by induction:}`

`\text{Test that the statement holds for n = 1}`


`LHS = 4(4 + 2) = 4(6) = 24`

`RHS = (4(4 + 1)(8 + 7))/(6) = (20 \cdot 15)/(6) = (300)/(6) \\eq 24`

This means that the statement is false, and thus, cannot be proven by induction.