Problem 59 - College Algebra

Question

Express each sum using summation notation. Use a lower limit of summation of your choice and \(k\) for the index of summation. $$a+(a+d)+(a+2 d)+\dots+(a+n d)$$

Step-by-Step Solution

Answer

The sum can be expressed in summation notation as \(\sum_{k=0}^{n} (a+kd)\).

Step 1: Identify the pattern in the sequence

In the given sequence \(a+(a+d)+(a+2 d)+\dots+(a+n d)\), the first term is 'a'. Successive terms are incrementing by the common difference 'd'.

Step 2: Express in summation notation

Using summation notation, the series can be written as \(\sum_{k=0}^{n} (a+kd)\). Here, 'k' is the index of summation which starts from 0 and goes up to 'n'. The summand '(a+kd)' represents the terms in the series, with 'k' multiplied by the common difference 'd' and 'a' being the first term.