Question

What is the value of the sum 5+9+...+(4n-3) if the equation (2n+1) (n-1) holds for n ≥ 2?

OPTION 1: n^2 - 1
OPTION 2: n^2 + 1
OPTION 3: 2n^2 - 1
OPTION 4: 2n^2 + 1

Answer 1

The value of the series sum 5+9+...+(4n-3), given the equality (2n+1)(n-1), by using a pairing strategy is shown to be 2n². This result is derived by transforming the series into 'n' terms of 'n', each pair summing to 2n.

Step-by-step explanation:

The question asks for the value of the series sum 5+9+...+(4n-3), given the equation (2n+1)(n-1). This equation simplifies to n² - 1, by expansion, so it seems that we are looking for a series that sums up to this expression or related to it. In considering the series provided in the question, we see that the sum is actually an arithmetic series, where each term increases by a constant difference of 4. However, there is a more efficient way to find its sum by using a trick that involves pairing terms from opposite ends of the series.

The trick essentially involves converting the sequence into 'n' terms of 'n'. By strategically pairing and rearranging terms, as mentioned in the reference material, and using the fact that there are n terms in the series, one can derive that the sum is equal to 2n². This pairing shows that each pair sums up to 2n, and since there are n such pairs, the total sum is 2n×n, which is 2n².