Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.4.6 + 5.7 + 6.8 + ... + 4n( 4n + 2)

Solution:

It is given that

4.6 + 5.7 + 6.8 + ... + 4n( 4n + 2) = [4 (4n + 1) (8n + 7)]/6

When n = 1

LHS = 4n(4n+2)

= 4 × 6 = 24

RHS = [4 (4n + 1) (8n + 7)]/6

= [4 × 5 × 15]/6

= 300/6

= 50

Here LHS ≠ RHS for n = 1 the series is untrue

The RHS should be replaced by 4 (n + 1) (n + 2) (4n - 3)³

Therefore, the statement is false for all positive integers n.

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.4.6 + 5.7 + 6.8 + ... + 4n( 4n + 2)

Summary:

Using mathematical induction the statement 4.6 + 5.7 + 6.8 + ... + 4n( 4n + 2) is false for all positive integers n.