How can I use summation notation to write the series 2, 4, 6, 8 for 10 terms?

To express the series 2, 4, 6, 8 in summation notation for the first 10 terms, we start by observing a pattern in the series. Each term can be represented as an arithmetic sequence where the first term (a) is 2 and the common difference (d) is also 2.

The n-th term of an arithmetic sequence can be calculated using the formula:

T_n = a + (n – 1)d

Plugging in the values:

T_n = 2 + (n – 1) * 2 = 2n

Thus, the series can be represented as:

2 + 4 + 6 + … + 20

In summation notation, this can be concisely written as:

∑_n=1¹⁰ 2n

This notation indicates that you should sum the values of 2n for n ranging from 1 to 10, which indeed gives you the first 10 terms of the series (2, 4, 6, 8, 10, 12, 14, 16, 18, 20).