Question

An arithmetic series is defined by the formula an = 2n + 1.

a) What is the first term of the series?
b) Write an expression for the sum of the first n terms of the series.

Answer 1

a) The first term of the series is found by plugging in n = 1 into the formula: \(a_1 = 2 \times 1 + 1 = 3\).

b) The sum of the first n terms of an arithmetic series can be expressed using the formula \(S_n = \frac{n}{2} [2a_1 + (n-1)d]\), where \(a_1\) is the first term and \(d\) is the common difference. In this case, the common difference is 2 (since \(a_{n+1} - a_n = (2(n+1) + 1) - (2n + 1) = 2\)), and the first term \(a_1\) is 3. Thus, the expression for the sum of the first n terms is:

\[S_n = \frac{n}{2} [2 \times 3 + (n-1) \times 2]\]