In sequences and series:::
I don't understand the difference between (for arthemtic series) the equations
1) sn= t/2(ti-tn)
2) (n* n-1)/2
where would you use each one and how are they different?
In sequences and series, both equations you mentioned are used to find the sum of an arithmetic series. However, these equations differ in terms of how the sum is calculated and the variables involved.
1) The formula "sn = (t/2)(ti - tn)" is used to find the sum of an arithmetic series when you have the first term (ti), the last term (tn), and the total number of terms (t). This formula is known as the "sum of arithmetic series formula" and can be applied when you have these specific values available.
2) On the other hand, the formula "(n * (n - 1))/2" is used to find the sum of an arithmetic series by using the number of terms (n) alone. This formula is derived from a simple pattern in arithmetic series, which shows that the sum of an arithmetic series is equal to half the product of the number of terms and the sum of the first and last term. Since the sum of the first and last term in an arithmetic series is equal to (n * (n - 1)), this formula can be simplified to (n * (n - 1))/2 to find the sum.
To summarize:
- Use "sn = (t/2)(ti - tn)" when you have the first term, last term, and total number of terms.
- Use "(n * (n - 1))/2" when you only have the number of terms.
It is important to note that both formulas give you the same result, which is the sum of the arithmetic series. The choice of which equation to use depends on the information you have available for the specific arithmetic series problem you are trying to solve.