Question

Identify the correct steps used to prove the formula Σnj=1(aj-aj-1)=an-a0, where an is a sequence of real numbers?

1) Use mathematical induction
2) Use the formula for the sum of an arithmetic series
3) Use the formula for the sum of a geometric series
4) Use the formula for the sum of a finite sequence

Answer 1

To prove the formula Σnj=1(aj-aj-1)=an-a0, where an is a sequence of real numbers, you can use mathematical induction, the formula for the sum of an arithmetic series, and the formula for the sum of a finite sequence.

Step-by-step explanation:

To prove the formula Σnj=1(aj-aj-1)=an-a0, where an is a sequence of real numbers, you can use the following steps:

1. Use mathematical induction: Start by proving the formula for n=1. Then assume it holds for n=k and prove it for n=k+1. This will establish the formula for all positive integers.
2. Use the formula for the sum of an arithmetic series: This formula states that the sum of an arithmetic series with n terms is given by Sn = (n/2)(a1 + an), where a1 is the first term and an is the nth term. Apply this formula to both sides of the equation Σnj=1(aj-aj-1)=an-a0 to simplify and prove the formula.
3. Use the formula for the sum of a finite sequence: This formula states that the sum of a finite sequence is given by Sn = an - a0. Apply this formula to both sides of the equation Σnj=1(aj-aj-1)=an-a0 to simplify and prove the formula.