Sequences and Series
Sequences and Series
Sequences and Series
𝑆 = 𝑆1 + 𝑆2 + ⋯ + 𝑆𝑛
Example 1
Write the first 4 terms of the sequence defined by
𝑎𝑛 = 3𝑛 − 2
Answer: 6,2,-2,-6 and -10 are the first 5 terms of the given
sequence.
Example 3
Find the general term for the sequence 2,7,12,…
𝑎𝑛 = 5𝑛 − 3
Checking:
𝑎1 = 5 1 − 3 = 2
𝑎1 = 5 2 − 3 = 7
𝑎1 = 5 3 − 3 = 12
Two Types of Sequence
I. Arithmetic Sequence
- There is a common difference between
successive terms. If any term is subtracted from
the next term, the result is always the same,
and this number is called common difference.
- In general arithmetic sequence whose first term
is 𝑎1 and whose common difference is 𝑑, the
𝑛𝑡ℎ term is therefore
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
Example 1
Find the 15th term of an arithmetic
sequence, if 𝑎1 = 5 and 𝑑 = 4.
Solution:
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎15 = 5 + (15 − 1)(4)
Answer: 𝑎15 = 61
Example 2
Find the number of terms in the arithmetic sequence,
1 5 8 29
, , ,…,
3 6 6 6
1 29
Applying the formula with a= and 𝑎𝑛 =
3 6
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
29 1 1
= + 𝑛−1
6 3 2
29 1 1 1
= + 𝑛−
6 3 2 2
n= 10
The Sum of n terms of an Arithmetic Sequence
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
But 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
Substituting,
𝑛
𝑆𝑛 = 2𝑎1 + 𝑛 − 1 𝑑
2
Example 1
Find the sum of 15 terms of the sequence 3,5,7,9,…
Solution: The common difference is 𝑑 = 2, with 𝑎1=3 and n = 15. Applying the
second formula
𝑛
𝑆𝑛 = 2𝑎1 + 𝑛 − 1 𝑑
2
15
𝑆15 = 2(3) + 15 − 1 2
2
15
= 6 + 28
2
𝑆15 = 255