Hello,
Could you please clarify the solution to the following question, specifically how it arrives at the expressions 2P - 1 and n + 1/2?
If you have an even number of terms then you know that your sum is going to be negative as you can see below.
\underbrace{(1-2)}_{-1} + \underbrace{(3-4)}_{-1}+ \underbrace{(5-6)}_{-1}+ \ldots ((n - 1) - n)
Specifically, it’s just (-1) \cdot \text{number of pairs} = -1 \left( \frac{n}{2}\right)
Needless to say, we multiply by (-1) because each pair sums to -1
In your case, you should thus know that you have an odd number of terms because the given sum is positive.
In that case, you have: \underbrace{(1-2)}_{-1} + \underbrace{(3-4)}_{-1}+ \underbrace{(5-6)}_{-1}+ \ldots + (\underbrace{(n-2) - (n-1))}_{-1} + n
Here, it’s evident that you have \frac{n-1}{2} pairs (each pair having a sum of -1) and the last term (n) to get your “total sum”. Therefore, “total sum” is just (-1) \left(\frac{n-1}{2}\right) + n, which simplifies to \frac{n + 1}{2}.
I think the rest is trivial and you hopefully know how to conclude.
Hello,
Thank you for your explanation.
If we consider the sequence 1, -2, 3, -4, …, we can observe the following:
- The first term is 1.
- For the next terms, when we add -2 and 3, we get 1.
- Similarly, adding -4 and 5 also results in 1.
This pattern indicates that:
- With 1 term, the sum is 1.
- With 3 terms, the sum is 2.
- With 5 terms, the sum is 3.
The formula can be expressed as: if we take twice the sum and subtract 1, we get the nth term ( 2 * sum - 1 = nth term).
Given that the sum is 180, we can conclude that the nth term would be 359 according to this formula.
This was my solution after reviewing and solving the question again this time.