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NOTES: Implicit vs. Explicit Formulas
CONSIDER THIS: A pyramid of logs has 2 logs in the top row, 4 logs in the second row from the top, 6 logs in
the third row from the top, and so on, until there are 200 logs in the bottom row.
Write and interpret the first 10 terms of the sequence of numbers generated from the example.
Identify the pattern in the sequence of numbers.
Write the formula for the nth term of the sequence and use it to find the number of logs in the 76
th
row.
Compute the number of logs in the first 12 rows combined.
What is the total number of logs in the pyramid?
CONSIDER THE SEQUENCE: 1 , 4 , 7 , 10 , 13 , …
𝑛 = the term number (think of it as the term’s place in line)
𝑛
= the n
th
term
1
represents the FIRST term.
2
represents the SECOND term.
3
represents the THIRD term.
4
represents the FOURTH term, etc…
𝑛− 1
= the previous term
𝑛+ 1
= the next term
IMPLICIT FORMULA: requires knowing the previous term
EXPLICIT FORMULA: requires only knowing the desired n.
FILL IN THE CHART:
Find the first six terms for each sequence:
𝑘
𝑘
𝑘
2
𝑘
𝑘− 1
1
𝑘
𝑘− 1
2
1
𝑘
𝑘− 1
1
NOTES: ARITHMETIC SEQUENCES
ARITHMETIC, GEOMETRIC, or NEITHER?
An arithmetic sequence is one where a constant value is _______________ to each term to get the next term.
example: {5, 7, 9, 11, …}
A geometric sequence is one where a constant value is _______________by each term to get the next term.
example: {5, 10, 20, 40, …}
EXAMPLE: Determine whether each of the following sequences is arithmetic, geometric, or neither:
a. {
1
2
1
4
1
8
1
16
b. {9, - 1, - 11, - 21, ...}
c. {0, 1, 1, 2, 3, 5, 8, 13, 21,...}
ARITHMETIC SEQUENCES:
A sequence is arithmetic if there exists a number d , called the common ________________________, such
that for 𝑑 = 𝑎
𝑛
𝑛− 1
for 𝑛 ≥ 2.
In other words, if we start with a particular first term, and then add the same number successively, we
obtain an arithmetic sequence.
Example: Write an explicit formula for the sequence {10, 15, 20, 25, …}.
In general, the explicit formula for an arithmetic sequence is given by _________________________________________.
NOTES: Arithmetic Series
What is an arithmetic SERIES? --the ______ of an indicated number of terms of a sequence.
Arithmetic Sequence: Arithmetic Series:
The sum of a finite arithmetic sequence with common difference d is 𝑆
𝑛
𝑛
2
1
𝑛
Example: Find the sum of the first 15 terms of the sequence { 1 , 5 , 9 , 13 , … }.
NOTE: This is called the 15
th
partial sum of the sequence.
Example: Find the sum of the first 100 terms of the sequence {-18, - 13, - 8, - 3, 2,…}.
NOTE: This is called the 100
th
partial sum of the sequence.
Example: Given the sum of the first 20 terms of a sequence that starts with 5 is 220, find the 20
th
term.
Example: Given the sum of the first 15 terms of an arithmetic sequence is 165 and the first term is − 3 ,
find…
the common difference.
the 15
th
term.
the explicit formula for the sequence.
the sum of the first 20 terms of the sequence.
Example: A corner section of a stadium has 14 seats along the front row and adds one more seat to each successive
row. If the top row has 35 seats, how many seats are in that section?
