4.03 Arithmetic series

Describe the difference between a sequence and a series.

Find the sum of the first 15 terms of the following arithmetic series:

Find the sum of the 14 terms in the arithmetic series 13 + \ldots + \dfrac{13}{2}.

Consider the arithmetic series - 9 - 5-1 + \ldots + 59 .

Consider the sum of the odd natural numbers: 1 + 3 + 5 + \ldots Show that the sum of the first n odd natural numbers is a perfect square.

Consider the arithmetic sequence 4, -1, -6, \ldots

For each of the following, find the sum of the first 10 terms of the arithmetic sequence:

The first term of an arithmetic sequence is - 5 and the sixth term is - 45.

Find the sum of the first 20 terms of the arithmetic series defined by T_1 = \pi and T_{20} = 20 \pi.

Find the sum of the first 40 terms of the following arithmetic sequences:

In an arithmetic progression, the sum of the first n terms is denoted by S_n. If S_{20} = 670, and S_{19} = 608, find T_{20}.

The first term of an arithmetic sequence is 7, the common difference is 3 and the sum of the first n terms is 205. Find the value of n.

For each of the following:

The sum of the first n terms of an arithmetic progression is S_n = 3 n^{2} - 8 n.

Consider an arithmetic progression in which the terms are increasing. The first of three consecutive terms in the progression is y. The sum of these three terms is - 12. The product of the three terms is 80.

State the value of the three consecutive terms.

Consider the following and rewrite each series using summation notation:

Find the value of the following:

Show that:

\sum_{i=1}^{n} \left(a_i + b_i\right) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i

Find the sum of all integers between 20 and 50, inclusive, that are divisible by 6.

A worker at a factory is stacking cylindrical-shaped pipes in layers. Each layer contains one pipe less than the layer below it. There are 4 pipes in the topmost layer, 5 pipes in the next layer, and so on. There are n layers in the stack.

Find the total number of pipes in the stack in terms of n.

Bart is learning to drive. His first lesson is 26 minutes long, and each subsequent lesson is 4 minutes longer than the lesson before.

If Bart reaches 9.6 total hours of driving on his nth lesson, find the value of n.

For a sprint training exercise, a number of balls are placed in a straight line. The first ball is 4 metres from the start, and there is a 3 metre distance between each of the remaining consecutive balls. There are n balls placed out in the line in total.

Pauline must run from the start, collect the nearest ball and run back to the start, depositing the ball into a box. Then she must run back to collect the next ball and bring it back to the start. She continues this until all n balls have been collected.

Pauline wants to runs 350 metres in total to collect and deposit all the balls. Solve for n, the number of balls that would need to be placed in a line.

Katrina starts training for a 4.5\text{ km} charity trail run by running every week for 28 weeks.

She runs 2\text{ km} of the course in the first week, and each week after that she runs 250 metres more than the previous week. She continues to increase the distance she runs until the week when she runs far enough to complete the course. Each week after that she completes the course without increase.

A fishing trawler spends several days netting crabs. On the first day, it nets 550\text{ kg} of crabs, on the next day it nets 538\text{ kg}. The amount netted continues to decrease by the same amount each day.