Arithmetic and Geometric

Find the Sum

Determine the sum of all the integers from 1 to 100.1 + 2 + 3 + … + 100 = ?

SUM MATHEMATICIAN

Carl Friedrich Gauss

Theorem:

The sum of the integers from 1 to n is given by the formula:

12

1 nnSn

Example/s:

1.Find the sum of all the integers from 1 to 25.

2.Find the sum of all the integers from 1 to 40.

Arithmetic Sequence

It is a sequence in which the difference between consecutive terms is constant and has the form:

Note: An arithmetic sequence

exhibit constant growth

dnadadaa 1,,2,, 1111

Arithmetic Sequence

Examples:1, 4, 7, 10, 13, …6, 11, 16, 21, 26, …14, 25, 36, 47, 58, …4, 2, 0, -2, -4, …-1, -7, -13, -19, -25, …

Arithmetic Sequence

General Term:

dnaan 11

Example/s:

For each arithmetic sequence, identify the first term a1, and the common difference.

4, 9, 14, 19, 24, …27, 20, 13, 6, -1, -8, …

Example/s:

Find the 50th term of the arithmetic sequence 2, 6, 10, 14, …

Example/s:

Find the 14th term of the arithmetic sequence 4, 7, 10, 13, …

Example/s:

Find the first term of the arithmetic sequence where the 3rd term is 13 and the 10th term is 62.

Example/s:

The 3rd term of an arithmetic sequence is 8, and the 16th term is 47. Find a1 and d.

Example/s:

The 3rd term of an arithmetic sequence is 8, and the 16th term is 47. Find a1 and d.

Example/s:

Find the general term for each of the following arithmetic sequences:

1, 5, 9, 13, …5, 2, -1, -4, …

Arithmetic Series

A series is an indicated sum of terms of a sequence. If the terms form an arithmetic sequence with first term a1 and common difference d, the indicated sum of terms is called an arithmetic series. The sum of the first n terms, represented as Sn, is

nnn aaaaaS 1321

Arithmetic Series

Let Sn = a1 + a2 + … + an be an arithmetic series then

21 n

n

aanS

Arithmetic Series

Let Sn = a1 + a2 + … + an be an arithmetic series with constant difference d, then:

2

12 1 dnanSn

Example/s:

Find the sum of the first 100 positive even numbers.

Example/s:

Find the sum of the first 15 terms of the arithmetic sequence 4, 7, 10, 13, …

Example/s:

Find the sum of the first 50 terms in the series 2 + 5 + 8 + 11 + …

Example/s:

Find the sum of all odd numbers between 7 and 433, inclusive.

Example/s:

A stack of telephone poles has 30 poles in the bottom row. There are 29 poles in the second row, 28 in the next row, and so on. How many poles are in the stack if there are 5 poles in the top row?

Geometric Sequence

(Or Geometric Progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a common multiplier. The common multiplier is called the common ratio of the sequence.

Geometric Sequence

Examples:1, 2, 4, 8, 16, …8, 4, 2, 1, ½, …3, 9, 27, 81, 243, …1, -4, 16, -64, 256, …

Geometric Sequence

General Term: The general term of a sequence is given by

Where a1 is the first term

and r is the common ratio

11

nn raa

Example/s:

For each geometric sequence find the common ratio.

3, 6, 12, 24, 48, …3, -6, 12, -24, 48, …

Example/s:

Find the 10th term of the geometric sequence 9, 3, 1, …

Example/s:

Find the 7th term of the geometric sequence 4, 20, 100, …

Example/s:

Find the general term for the geometric sequence 2, 4, 8, 16, …

Geometric Series

A geometric series is the indicated sum of a geometric sequence. The following are examples of geometric series:

1 + 2 + 4 + 8 + …8 + 4 + 2 + 1 + …1 + (-4) + 16 + (-64) + …

Geometric Series

The sum of the first n terms of a geometric sequence that has a first term a1 and a common ration r is given by:

.1any for ,

1

11

rr

raS

n

n

Example/s:

Find the sum of the first 7 terms of the geometric series 2 + 6 + 18 + …

Example/s:

Find the sum of the first 10 terms of the geometric series 3, -6, 12, -24, …

Example/s:

Find the arithmetic mean of 4 and 16.

Example/s:

Find the arithmetic mean of 3 and 21.

Example/s:

Find the arithmetic mean of 11 and 23.

Example/s:

Insert three arithmetic means between 3 and 11.

Example/s:

Insert two arithmetic means between 15 and 36.

Example/s:

Insert 3 arithmetic means between 52 and 40.

Example/s:

Find the geometric mean between 5 and 45.

Example/s:

Find the geometric mean of 3 and 48.

Example/s:

Insert two geometric means between 4 and 500.

Short Quiz 1:1. Find the sum of the first 40 terms of the arithmetic sequence 3, 7, 11, 15,….2. Find the sum of the first 50 odd numbers.3. An amphitheatre has 50 rows of seats with 30 seats in the first row, 32 in the second, 34 in the third, and so on. Find the total number of seats.4. Find the first 6 terms of the arithmetic sequence 13, 7,…and also the 300th term..