Where does one term end and the other begin?

[trigger352]

I need help determining the explicit formula and writing the series in sigma notation with a specified lower limit.

In both of them, I cannot tell where one term ends and the other begins, or what to do when the sign changes.

Series #1
$$- 2 - 5 - 8 - 11 - 14 - 17 - 20$$; lower limit = $$3$$
Series #2
$$4.4 + 2.4 + 0.4 - 1.6 - 3.6 - 5.6 - 7.6$$; lower limit = $$2$$

 

[chroot]

Each term is a number, either positive or negative. Since this is a series, you are implicitly adding all the terms. The terms which are negative get rid of the plus sign before them, per convention.

- Warren

 

[thepatientmental]

The explicit formula for Series #1 is given by a_n = -3n + 1, where n is the term number. To write this series in sigma notation with a lower limit of 3, we can use the following expression:

∑(n=3 to ∞) (-3n + 1)

Similarly, the explicit formula for Series #2 is a_n = 6 - 2n, where n is the term number. To write this series in sigma notation with a lower limit of 2, we can use the following expression:

∑(n=2 to ∞) (6 - 2n)

In terms of where one term ends and the other begins, it is important to note that in both series, the terms are being subtracted from each other. So in Series #1, the first term is -2, the second term is -5, and the difference between these two terms is -3. This pattern continues for the rest of the series. Similarly, in Series #2, the first term is 4.4, the second term is 2.4, and the difference between these two terms is -2. This pattern also continues for the rest of the series.

Therefore, in both series, the sign change indicates the end of one term and the beginning of the next. To find the sum of these series, we can use the formula for a finite arithmetic series: S_n = (n/2)(a_1 + a_n), where n is the number of terms and a_1 and a_n are the first and last terms, respectively.