Arithmetic and Geometric
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, …
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 first term of the arithmetic sequence where the 3rd term is 13 and the 10th term is 62.
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 with constant difference d, then:
2
12 1 dnanSn
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, …
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
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..