1. Calculate 1+3+5+...+(2n-1). (Use any methods).
Answer (1)
To calculate the sum of the series [tex]$$1 + 3 + 5 + \cdots + (2n - 1)$$[/tex], you are essentially adding the first [tex]$$n$$[/tex] odd numbers.The formula for the sum of the first [tex]$$n$$[/tex] odd numbers is: [tex]S_n = n^2[/tex]This means the sum is simply the square of the number of terms.ExplanationThe series 1, 3, 5, 7, [tex]\ldots, (2n - 1)[/tex] is an arithmetic sequence where the first term [tex]$$a = 1$$[/tex] and the common difference [tex]$$d = 2$$[/tex].The number of terms is [tex]n[/tex].The sum of [tex]n[/tex] terms of an arithmetic sequence is: [tex]S_n = \frac{n}{2} \times (a + l)[/tex]where [tex]l[/tex] is the last term, which is [tex]$$2n - 1$$[/tex].Substituting values: [tex]S_n = \frac{n}{2} \times (1 + 2n - 1) = \frac{n}{2} \times 2n = n^2[/tex]ExampleIf [tex]n = 5[/tex], the sum is: [tex]1 + 3 + 5 + 7 + 9 = 5^2 = 25[/tex]So, the sum of the first 5 odd numbers is 25.
