Series construction

When it comes to constructing a series, there are various methods that one can use. Two of the most common methods are arithmetic and geometric series.

Arithmetic series

Arithmetic series is a sequence of numbers in which each term is the sum of the previous term and a constant value called the common difference. For example, the series:

2, 5, 8, 11, 14, ...

can be expressed as an arithmetic series with a common difference of 3.

To find the sum of an arithmetic series, we use the formula:

Sn = n/2[2a + (n-1)d]

Where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.

For example, if we want to find the sum of the first 10 terms of the series:

3, 7, 11, 15, 19, ...

We can use the formula:

S10 = 10/2[2(3) + (10-1)4] = 10/2[6 + 36] = 210

Therefore, the sum of the first 10 terms of the series is 210.

Geometric series

Geometric series is a sequence of numbers in which each term is the product of the previous term and a constant value called the common ratio. For example, the series:

1, 2, 4, 8, 16, ...

can be expressed as a geometric series with a common ratio of 2.

To find the sum of a geometric series, we use the formula:

Sn = a(1 - r^n) / (1 - r)

Where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

For example, if we want to find the sum of the first 5 terms of the series:

3, 6, 12, 24, 48, ...

We can use the formula:

S5 = 3(1 - 2^5) / (1 - 2) = 3(-31) / (-1) = 93

Therefore, the sum of the first 5 terms of the series is 93.

In conclusion, arithmetic and geometric series are two common methods for constructing series. By using the formulas for these series, we can find the sum of a given number of terms in the series.

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