In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $a+(a+d)+(a+2d)+\cdots +(a+nd)$

Verified step by step guidance

1. Identify the pattern in the sum: the terms are a, (a + d), (a + 2d), ..., (a + nd), which shows an arithmetic sequence where each term increases by d.
2. Choose the index of summation as k, starting from 0 to n, because the first term corresponds to k = 0 and the last term corresponds to k = n.
3. Express the general term of the sequence using k: the k-th term is $a+kd$.
4. Write the summation notation by summing the general term from k = 0 to k = n: $\sum$_k=0ⁿ (a + kd).
5. This summation notation compactly represents the entire sum $a\; +\; (a\; +\; d)\; +\; (a\; +\; 2d)\; +\; \dots \; +\; (a\; +\; nd)$.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference, d, to the previous term. In the given sum, the terms follow this pattern: a, a+d, a+2d, ..., a+nd, which is the foundation for expressing the sum in summation notation.

Summation Notation

Summation notation uses the sigma symbol (∑) to represent the sum of a sequence of terms. It includes an index of summation, a lower limit, and an upper limit, allowing a compact expression of long sums like the arithmetic sequence given in the problem.

Index of Summation and Limits

The index of summation, often denoted by k, is a variable that represents each term's position in the sequence. The lower and upper limits define the range of terms to be summed. Choosing appropriate limits and expressing each term in terms of k is essential to correctly write the sum in summation notation.