FLC
Ch 12: Part II
Math 370 Precalculus
Sec 12.4: Mathematical Induction
Mathematical induction is a method used to prove that statements involving natural numbers are true
for all 𝑛𝜖ℕ.
Important:
It is extremely crucial to show each step when constructing proofs, including proofs by
induction. When writing formal proofs, written statements are just as critical as algebraic ones. Proofs
need to be rigorous and steps that are "obvious" should NOT be left to the reader. If it is obvious to
you, show that it is. Proofs will be graded on rigor and clarity. Refer to your notes or the provided
handout ("Proof by Mathematical Induction") for acceptable proofs.
Theorem
The Principle of Mathematic Induction
Suppose that the following two conditions are satisfied with regard to a statement about natural
numbers:
CONDITION I: The statement is true for the natural number 1.
CONDITION II: If the statement is true for some natural number 𝑘, it is also true for the next
natural number 𝑘 + 1.
Then the statement is true for ∀𝑛 ∈ ℕ (for all natural numbers).
Example in book (dominos) and "always right" example.
Ex 1 Prove that 1 + 3 + 5 + ⋯ + 2𝑛 − 1 = 𝑛2 for all natural numbers 𝑛.
Ex 2 (#2) Prove that 1 + 5 + 9 + ⋯ + 4𝑛 − 3 = 𝑛 2𝑛 − 1 for all natural numbers 𝑛.
Page 1 of 5
FLC
Ch 12: Part II
Ex 3 (#19) Prove that 𝑛2 + 𝑛 is divisible by 2 for all natural numbers 𝑛.
Ex 4 (#7) Prove that 1 + 2 + 4 + ⋯ + 2𝑛−1 = 2𝑛 − 1 for all natural numbers 𝑛.
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FLC
Ch 12: Part II
Ex 5 (#20) Prove that 𝑛3 + 2𝑛 is divisible by 3 for all natural numbers 𝑛.
Sec 12.5: The Binomial Theorem
There are two ways of counting -- using permutations or combinations and both count the number of
ways to select 𝑘 distinct objects from a set of 𝑛 elements. When finding the 𝑘 permutations on the set,
we take order into account. Combinations are permutations where the order does not matter.
Defn
"𝑛 choose 𝑗"
If 𝑗 and 𝑛 are integers with 0 ≤ 𝑗 ≤ 𝑛, the symbol
𝑛
𝑗
𝑛
0
𝑛
is defined as 𝑗
=
𝑛!
𝑗 ! 𝑛 −𝑗 !
.
may also be denoted 𝑛𝐶𝑗.
Ex 6 Evaluate each.
(#5)
𝑛
𝑗
5
3
(#16)
𝑛
1
37
19
4
2
𝑛
𝑛−1
𝑛
𝑛
B
G
R
P
Page 3 of 5
FLC
Ch 12: Part II
How do we expand 𝑥 + 𝑎 𝑛 ?
See if you notice any patterns
𝑥+𝑎
0
=1
1 term
𝑥+𝑎
1
=𝑥+𝑎
2 terms
𝑥+𝑎
2
= 𝑥 2 + 2𝑎𝑥 + 𝑎2
3 terms
𝑥+𝑎
3
= 𝑥 3 + 3𝑎𝑥 2 + 3𝑎2 𝑥 + 𝑎3
4 terms
𝑥+𝑎
4
= 𝑥 4 + 4𝑎𝑥 3 + 6𝑎2 𝑥 2 + 4𝑎3 𝑥 + 𝑎4
5 terms
𝑥+𝑎
5
= 𝑥 5 + 5𝑎𝑥 4 + 10𝑎2 𝑥 3 + 10𝑎3 𝑥 2 + 5𝑎4 𝑥 + 𝑎5
6 terms
Patterns:
1) # of terms: 𝑥 + 𝑎
𝑛
has ________ terms
2) first and last term of 𝑥 + 𝑎
𝑛
are: ________ and ________, respectively
3) powers of 𝑥 and 𝑎: powers of 𝑥 ______________ by 1 whereas the powers of 𝑎 ______________ by 1
4) sum of exponents for each term in expansion: add up to ______
What about the coefficients?
The pattern can be found in Pascal’s Triangle.
Pascal’s Triangle
The Binomial Theorem
Let 𝑥 and 𝑎 be real numbers. For any given natural number 𝑛, we have
𝑥+𝑎
𝑛
=
𝑛
0
𝑛
𝑥 +
𝑛
Note: The symbol 𝑗
𝑛
1
𝑎𝑥
𝑛−1
+ ⋯+
𝑛
𝑗
𝑗
𝑎 𝑥
𝑛−𝑗
+ ⋯+
𝑛
𝑛
𝑛
𝑎 =
𝑛
𝑗 =0
𝑛
𝑗
𝑎 𝑗 𝑥 𝑛−𝑗
is called a binomial coefficient.
The term containing 𝑥 𝑗 is
𝑛
𝑎𝑛−𝑗 𝑥 𝑗
𝑛−𝑗
If 𝑛 and 𝑗 are integers with 1 ≤ 𝑗 ≤ 𝑛, then
𝑛
𝑛
𝑛+1
+
=
𝑗−1
𝑗
𝑗
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FLC
Ch 12: Part II
Ex 7 Expand each.
(#20)
𝑥+3 5
3𝑥 − 2
6
𝑥− 3
4
Ex 8 Find the indicated coefficient or term.
(#34) The coefficient of 𝑥 2 in the expansion of 2𝑥 − 3 9 .
(#38)
The sixth term in the expansion of 3𝑥 + 2 8 .
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