Charlotte is trying to prove the summation formula
She has checked that the basis is correct, has assumed that and is trying to show that
Charlotte knows that she needs to express in terms of her assumption for the , but she cannot quite remember the method. Determine which of the following is correct.
David has read in a textbook that David wants to prove this using induction.
First, he starts with the basis step substituting into each side of the equation. He calculates that the left-hand side, equals 1. Calculate the value of the right-hand side, and, hence, determine if the basis is true.
David assumes that the summation formula is true when giving him that For the induction step, he needs to show that Using the fact that substitute in David’s assumption and simplify the result to find an expression for
David then makes the following conclusion:
If our assumption is correct for , we have shown that the summation formula is correct when . Therefore, as we have shown that the summation formula is true when , by mathematical induction, the formula is true for all natural numbers .
Is David’s conclusion correct?